Andrew Heathwaite's MOS Investigations: Difference between revisions
Wikispaces>Andrew_Heathwaite **Imported revision 287859632 - Original comment: ** |
Wikispaces>Andrew_Heathwaite **Imported revision 288263754 - 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>2011-12- | : This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2011-12-23 03:47:15 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>288263754</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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While 37edo porcupine distinguishes the full variety of 1-steps and 2-steps, 22edo and 15edo porcupine do not. 22edo has L=dd and A=sd. 15edo has s=dd, L=sd, and A=Ld. This suggests more chromatic/enharmonic nuance is available in 37edo porcupine, while more ambiguity and "puns" are available in 22edo and 15edo porcupine. | While 37edo porcupine distinguishes the full variety of 1-steps and 2-steps, 22edo and 15edo porcupine do not. 22edo has L=dd and A=sd. 15edo has s=dd, L=sd, and A=Ld. This suggests more chromatic/enharmonic nuance is available in 37edo porcupine, while more ambiguity and "puns" are available in 22edo and 15edo porcupine. | ||
==L/s ratio and consequences on MODMOS steps== | |||
Different L/s ratios produce different relationships between the steps of the above scales. | |||
Case A: L/s<2. The chroma is smaller than the s step (c<s). | |||
Case B: L/s=2, ie. L=2s. The chroma and the small step are the same (c=s). | |||
Case C: L/s>2. The chroma is larger than the s step (c>s). | |||
Case A1: 2<L/s<3/2. The diminished step (s-c) is smaller than the chroma (d<c). | |||
Case A2: L/s=3/2. The diminished step and the chroma are the same (d=c). | |||
Case A3: 3/2<L/s<1. The diminished step is larger than the chroma (d>c). | |||
For Case B, there is no diminished step. Removing a chroma from an s step causes it to vanish. (d=0). | |||
For Case C, assuming the diminished step is s-c, the diminished step is negative! Mike Battaglia, on FB, suggests we say that d is |s-c|. In this case, the diminished step is always smaller than the chroma. (d<c). | |||
In all cases, A (L+c) is larger than s, c, or d. | |||
In "An Approach to Chromatic/Enharmonic MODMOS scales" above, the first four examples satisfied Case A3, and so in each example A<L<s<d<c. A/__**L/s**__/d/c for each was: Sensi[8] in 46edo: 9/__**7/5**__/3/2; Miracle[10] in 72edo: 11/__**9/7**__/5/2; Porcupine[7] in 37edo: 9/__**7/5**__/3/2; Porcupine[7] in 22edo: 5/__**4/3**__/2/1. Notice the L/s ratio (bolded and underlined above) is between 1 and 3/2 in all cases. The exception is Porcupine[7] in 15edo, with L/s=3/2, and the A/L/s/d/c is 4/__**3/2**__/1/1. The 15edo example, which is Case A2 above, is the only one which doesn't distinguish all five of these primary steps. | |||
My attraction to MODMOS scales in this range has to do with the fact that the smallest interval never appears melodically in a MODMOS, but only appears as the //difference// between some interval and one of its alterations. It seems to me this allows the greatest variety of potential MODMOS scales while assuring that no melodic steps appear that are "too small". Alternatively, it could mean that //when// the tiny chroma appears in a melody, it is signifying a change of mode. The limited range of L/s also assures that L and s are relatively close to the same size, and the original MOS and probably many of its MODMOSes has some of the very general melodic character of an edo. | |||
Since we know the borders of this region to be L/s=1/1 and L/s=3/2, we know that the simplest L/s ratio with L and s whole numbers is (1+3)/(1+2)=4/3. This is merely a "freshman sum" -- the sum of the two numerators over the sum of the two denominators. We can expand this range by continuing to do freshman sums: | |||
===> ===> ===> L and s closer to equal ===> ===> ===> | |||
<=== <=== <=== c and d closer to equal <=== <=== <=== | |||
|| 3/2 || || || || || || || || || || || || || || || || 1/1 || | |||
|| || || || || || || || || 4/3 || || || || || || || || || | |||
|| || || || || 7/5 || || || || || || || || 5/4 || || || || || | |||
|| || || 10/7 || || || || 11/8 || || || || 9/7 || || || || 6/5 || || || | |||
|| || 13/9 || || 17/12 || || 18/13 || || 15/11 || || 13/10 || || 14/11 || || 11/9 || || 7/6 || || | |||
Notice that on each far end of the spectrum there is a different ambiguity. Moving toward L/s=1/1, L and s become closer to equal; moving toward L/s=3/2, c and d become closer to equal. | |||
==Expanding on "Maximal Evenness"== | ==Expanding on "Maximal Evenness"== | ||
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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: | <!-- ws:start:WikiTextLocalImageRule:1062:&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:1062 --><!-- ws:start:WikiTextLocalImageRule:1063:&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:1063 --><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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While 37edo porcupine distinguishes the full variety of 1-steps and 2-steps, 22edo and 15edo porcupine do not. 22edo has L=dd and A=sd. 15edo has s=dd, L=sd, and A=Ld. This suggests more chromatic/enharmonic nuance is available in 37edo porcupine, while more ambiguity and &quot;puns&quot; are available in 22edo and 15edo porcupine.<br /> | While 37edo porcupine distinguishes the full variety of 1-steps and 2-steps, 22edo and 15edo porcupine do not. 22edo has L=dd and A=sd. 15edo has s=dd, L=sd, and A=Ld. This suggests more chromatic/enharmonic nuance is available in 37edo porcupine, while more ambiguity and &quot;puns&quot; are available in 22edo and 15edo porcupine.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="x-Expanding on &quot;Maximal Evenness&quot;"></a><!-- ws:end:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="x-L/s ratio and consequences on MODMOS steps"></a><!-- ws:end:WikiTextHeadingRule:4 -->L/s ratio and consequences on MODMOS steps</h2> | ||
Different L/s ratios produce different relationships between the steps of the above scales.<br /> | |||
<br /> | |||
Case A: L/s&lt;2. The chroma is smaller than the s step (c&lt;s).<br /> | |||
Case B: L/s=2, ie. L=2s. The chroma and the small step are the same (c=s).<br /> | |||
Case C: L/s&gt;2. The chroma is larger than the s step (c&gt;s).<br /> | |||
<br /> | |||
Case A1: 2&lt;L/s&lt;3/2. The diminished step (s-c) is smaller than the chroma (d&lt;c).<br /> | |||
Case A2: L/s=3/2. The diminished step and the chroma are the same (d=c).<br /> | |||
Case A3: 3/2&lt;L/s&lt;1. The diminished step is larger than the chroma (d&gt;c).<br /> | |||
<br /> | |||
For Case B, there is no diminished step. Removing a chroma from an s step causes it to vanish. (d=0).<br /> | |||
For Case C, assuming the diminished step is s-c, the diminished step is negative! Mike Battaglia, on FB, suggests we say that d is |s-c|. In this case, the diminished step is always smaller than the chroma. (d&lt;c).<br /> | |||
In all cases, A (L+c) is larger than s, c, or d.<br /> | |||
<br /> | |||
In &quot;An Approach to Chromatic/Enharmonic MODMOS scales&quot; above, the first four examples satisfied Case A3, and so in each example A&lt;L&lt;s&lt;d&lt;c. A/<u><strong>L/s</strong></u>/d/c for each was: Sensi[8] in 46edo: 9/<u><strong>7/5</strong></u>/3/2; Miracle[10] in 72edo: 11/<u><strong>9/7</strong></u>/5/2; Porcupine[7] in 37edo: 9/<u><strong>7/5</strong></u>/3/2; Porcupine[7] in 22edo: 5/<u><strong>4/3</strong></u>/2/1. Notice the L/s ratio (bolded and underlined above) is between 1 and 3/2 in all cases. The exception is Porcupine[7] in 15edo, with L/s=3/2, and the A/L/s/d/c is 4/<u><strong>3/2</strong></u>/1/1. The 15edo example, which is Case A2 above, is the only one which doesn't distinguish all five of these primary steps.<br /> | |||
<br /> | |||
My attraction to MODMOS scales in this range has to do with the fact that the smallest interval never appears melodically in a MODMOS, but only appears as the <em>difference</em> between some interval and one of its alterations. It seems to me this allows the greatest variety of potential MODMOS scales while assuring that no melodic steps appear that are &quot;too small&quot;. Alternatively, it could mean that <em>when</em> the tiny chroma appears in a melody, it is signifying a change of mode. The limited range of L/s also assures that L and s are relatively close to the same size, and the original MOS and probably many of its MODMOSes has some of the very general melodic character of an edo.<br /> | |||
<br /> | |||
Since we know the borders of this region to be L/s=1/1 and L/s=3/2, we know that the simplest L/s ratio with L and s whole numbers is (1+3)/(1+2)=4/3. This is merely a &quot;freshman sum&quot; -- the sum of the two numerators over the sum of the two denominators. We can expand this range by continuing to do freshman sums:<br /> | |||
<br /> | |||
===&gt; ===&gt; ===&gt; L and s closer to equal ===&gt; ===&gt; ===&gt;<br /> | |||
&lt;=== &lt;=== &lt;=== c and d closer to equal &lt;=== &lt;=== &lt;===<br /> | |||
<table class="wiki_table"> | |||
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<td>3/2<br /> | |||
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<td>1/1<br /> | |||
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<td>4/3<br /> | |||
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<td>7/5<br /> | |||
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<td><br /> | |||
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<td>5/4<br /> | |||
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<td>10/7<br /> | |||
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<td><br /> | |||
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<td>11/8<br /> | |||
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<td><br /> | |||
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<td>9/7<br /> | |||
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<td>6/5<br /> | |||
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<td><br /> | |||
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<td>13/9<br /> | |||
</td> | |||
<td><br /> | |||
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<td>17/12<br /> | |||
</td> | |||
<td><br /> | |||
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<td>18/13<br /> | |||
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<td>15/11<br /> | |||
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<td>13/10<br /> | |||
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<td>14/11<br /> | |||
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<td>11/9<br /> | |||
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<td>7/6<br /> | |||
</td> | |||
<td><br /> | |||
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</tr> | |||
</table> | |||
<br /> | |||
Notice that on each far end of the spectrum there is a different ambiguity. Moving toward L/s=1/1, L and s become closer to equal; moving toward L/s=3/2, c and d become closer to equal.<br /> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc3"><a name="x-Expanding on &quot;Maximal Evenness&quot;"></a><!-- ws:end:WikiTextHeadingRule:6 -->Expanding on &quot;Maximal Evenness&quot;</h2> | |||
&quot;<a class="wiki_link" href="/Maximal%20Evenness">Maximal Evenness</a>&quot; (ME, aka &quot;Quasi-Equalness,&quot; QE) is a quality certain MOS scales within equal scales can have.<br /> | &quot;<a class="wiki_link" href="/Maximal%20Evenness">Maximal Evenness</a>&quot; (ME, aka &quot;Quasi-Equalness,&quot; QE) is a quality certain MOS scales within equal scales can have.<br /> | ||
<br /> | <br /> | ||
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So that's where I'm leaving this problem for now.<br /> | So that's where I'm leaving this problem for now.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:8:&lt;h2&gt; --><h2 id="toc4"><a name="x-MOS Scales with similar generators"></a><!-- ws:end:WikiTextHeadingRule:8 -->MOS Scales with similar generators</h2> | ||
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: | <!-- ws:start:WikiTextLocalImageRule:1064:&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:1064 --><br /> | ||
<!-- ws:start:WikiTextLocalImageRule: | <!-- ws:start:WikiTextLocalImageRule:1065:&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:1065 --><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 /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc5"><a name="x-Notes on Keenan Pepper's Diatonic-like MOS Scales"></a><!-- ws:end:WikiTextHeadingRule:10 -->Notes on Keenan Pepper's Diatonic-like MOS Scales</h2> | ||
<br /> | <br /> | ||
In the Xenharmonic Alliance Facebook Group, on Dec. 1, 2011, Keenan Pepper posted a short list of MOS scales, introducing them, '<span class="messageBody">The diatonic scale has both an extremely low average harmonic entropy, and also a very nearly maximum 'categorical channel capacity' (something I'm currently working on defining properly in terms of information theory - it basically means 'ability to tell different intervals and modes apart').</span>&quot;<br /> | In the Xenharmonic Alliance Facebook Group, on Dec. 1, 2011, Keenan Pepper posted a short list of MOS scales, introducing them, '<span class="messageBody">The diatonic scale has both an extremely low average harmonic entropy, and also a very nearly maximum 'categorical channel capacity' (something I'm currently working on defining properly in terms of information theory - it basically means 'ability to tell different intervals and modes apart').</span>&quot;<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:12:&lt;h1&gt; --><h1 id="toc6"><a name="Porcupine Temperament"></a><!-- ws:end:WikiTextHeadingRule:12 -->Porcupine Temperament</h1> | ||
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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>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 /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:14:&lt;h1&gt; --><h1 id="toc7"><a name="Porcupine Chromaticism"></a><!-- ws:end:WikiTextHeadingRule:14 -->Porcupine Chromaticism</h1> | ||
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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 /> | ||
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<!-- ws:start:WikiTextLocalImageRule: | <!-- ws:start:WikiTextLocalImageRule:1066:&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:1066 --><br /> | ||
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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 <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 /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:16:&lt;h2&gt; --><h2 id="toc8"><a name="Porcupine Chromaticism-Modes of Porcupine[7]"></a><!-- ws:end:WikiTextHeadingRule:16 -->Modes of Porcupine[7]</h2> | ||
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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 &quot;-3&quot; in the chain has a perfect fifth over the bass.<br /> | 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 &quot;-3&quot; in the chain has a perfect fifth over the bass.<br /> | ||
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<strong>4 3 3 3 3 3 3 .. -6 -5 -4 -3 -2 -1 0</strong><br /> | <strong>4 3 3 3 3 3 3 .. -6 -5 -4 -3 -2 -1 0</strong><br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:18:&lt;h2&gt; --><h2 id="toc9"><a name="Porcupine Chromaticism-Modes of Porcupine[7] that have one chromatic alteration"></a><!-- ws:end:WikiTextHeadingRule:18 -->Modes of Porcupine[7] that have one chromatic alteration</h2> | ||
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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.<br /> | 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.<br /> | ||
| Line 1,447: | Line 1,689: | ||
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 /> | 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 /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:20:&lt;h1&gt; --><h1 id="toc10"><a name="Orwell[9], meet Porcupine[7]"></a><!-- ws:end:WikiTextHeadingRule:20 -->Orwell[9], meet Porcupine[7]</h1> | ||
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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 &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 /> | 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 /> | ||
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Not even close!<br /> | Not even close!<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:22:&lt;h1&gt; --><h1 id="toc11"><a name="Names for steps"></a><!-- ws:end:WikiTextHeadingRule:22 -->Names for steps</h1> | ||
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This is getting silly! We need better names.....<br /> | This is getting silly! We need better names.....<br /> | ||