Andrew Heathwaite's MOS Investigations
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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= I'm going to zoom in on [[Porcupine|Porcupine Temperament]], 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: * 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. * Mike Battaglia writes about how 81/80 is "tempered in" to 25/24, making it melodically useful instead of an "irritating mystery interval" which "introduces pitch drift". * 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>." * 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.) * 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>" =Porcupine Chromaticism= 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 [[140edo]], which is arguably an optimal tuning for Porcupine. Take a look: ...coming soon... ==Modes of Porcupine[7]== The following modes are given in steps of 22edo. To generalize, 3 stands for the neutral third generating interval, and 4 stands for the wide major second. On the right is a contiguous chain of 7 tones separated by 6 iterations of the Porcupine generator. 3 3 3 3 3 3 4 .. 0 1 2 3 4 5 6 3 3 3 3 3 4 3 .. -1 0 1 2 3 4 5 3 3 3 3 4 3 3 .. -2 -1 0 1 2 3 4 3 3 3 4 3 3 3 .. -3 -2 -1 0 1 2 3 3 3 4 3 3 3 3 .. -4 -3 -2 -1 0 1 2 3 4 3 3 3 3 3 .. -5 -4 -3 -2 -1 0 1 4 3 3 3 3 3 3 .. -6 -5 -4 -3 -2 -1 0 ==Chromatically-Altered Modes of Porcupine[7]== The following list includes all the modes that can be generated by shifting one tone of Porcupine[7] by one quartertone interval (chroma), which is one degree in 22edo. 2 4 3 3 3 3 4 .. 0 _ 2 3 4 5 6 _ 8 4 3 3 3 3 4 2 .. -8 _ -6 -5 -4 -3 -2 _ 0 3 3 3 3 4 2 4 .. -2 _ 0 1 2 3 4 _ 6 3 3 3 4 2 4 3 .. -3 _ -1 0 1 2 3 _ 5 3 3 4 2 4 3 3 .. -4 _ -2 -1 0 1 2 _ 4 3 4 2 4 3 3 3 .. -5 _ -3 -2 -1 0 1 _ 3 4 2 4 3 3 3 3 .. -6 _ -4 -3 -2 -1 0 _ 2 2 3 4 3 3 3 4 .. 0 _ _ 3 4 5 6 _ 8 9 3 4 3 3 3 4 2 .. -8 _ _ -5 -4 -3 -2 _ 0 1 4 3 3 3 4 2 3 .. -9 _ _ -6 -5 -4 -3 _ -1 0 3 3 3 4 2 3 4 .. -3 _ _ 0 1 2 3 _ 5 6 3 3 4 2 3 4 3 .. -4 _ _ -1 0 1 2 _ 4 5 3 4 2 3 4 3 3 .. -5 _ _ -2 -1 0 1 _ 3 4 4 2 3 4 3 3 3 .. -6 _ _ -3 -2 -1 0 _ 1 2 2 4 3 3 3 4 3 .. -1 0 _ 2 3 4 5 _ _ 8 4 3 3 3 4 3 2 .. -9 -8 _ -6 -5 -4 -3 _ _ 0 3 3 3 4 3 2 4 .. -3 -2 _ 0 1 2 3 _ _ 6 3 3 4 3 2 4 3 .. -4 -3 _ -1 0 1 2 _ _ 5 3 4 3 2 4 3 3 .. -5 -4 _ -2 -1 0 1 _ _ 4 4 3 2 4 3 3 3 .. -6 -5 _ -3 -2 -1 0 _ _ 3 3 2 4 3 3 3 4 .. 0 1 _ 3 4 5 6 _ _ 9 2 3 3 4 3 3 4 .. 0 _ _ _ 4 5 6 _ 8 9 10 3 3 4 3 3 4 2 .. -8 _ _ _ -4 -3 -2 _ 0 1 2 3 4 3 3 4 2 3 .. -9 _ _ _ -5 -4 -3 _ -1 0 1 4 3 3 4 2 3 3 .. -10 _ _ _ -6 -5 -4 _ -2 -1 0 3 3 4 2 3 3 4 .. -4 _ _ _ 0 1 2 _ 4 5 6 3 4 2 3 3 4 3 .. -5 _ _ _ -1 0 1 _ 3 4 5 4 2 3 3 4 3 3 .. -6 _ _ _ -2 -1 0 _ 2 3 4 2 4 3 3 4 3 3 .. -2 -1 0 _ 2 3 4 _ _ _ 8 4 3 3 4 3 3 2 .. -10 -9 -8 _ -6 -5 -4 _ _ _ 0 3 3 4 3 3 2 4 .. -4 -3 -2 _ 0 1 2 _ _ _ 6 3 4 3 3 2 4 3 .. -5 -4 -3 _ -1 0 1 _ _ _ 5 4 3 3 2 4 3 3 .. -6 -5 -4 _ -2 -1 0 _ _ _ 4 3 3 2 4 3 3 4 .. 0 1 2 _ 4 5 6 _ _ _ 10 3 2 4 3 3 4 3 .. -1 0 1 _ 3 4 5 _ _ _ 9 2 3 3 3 4 3 4 .. 0 _ _ _ _ 5 6 _ 8 9 10 11 3 3 3 4 3 4 2 .. -8 _ _ _ _ -3 -2 _ 0 1 2 3 3 3 4 3 4 2 3 .. -9 _ _ _ _ -4 -3 _ -1 0 1 2 3 4 3 4 2 3 3 .. -10 _ _ _ _ -5 -4 _ -2 -1 0 1 4 3 4 2 3 3 3 .. -11 _ _ _ _ -6 -5 _ -3 -2 -1 0 3 4 2 3 3 3 4 .. -5 _ _ _ _ 0 1 _ 3 4 5 6 4 2 3 3 3 4 3 .. -6 _ _ _ _ -1 0 _ 2 3 4 5 2 4 3 4 3 3 3 .. -3 -2 -1 0 _ 2 3 _ _ _ _ 8 4 3 4 3 3 3 2 .. -11 -10 -9 -8 _ -6 -5 _ _ _ _ 0 3 4 3 3 3 2 4 .. -5 -4 -3 -2 _ 0 1 _ _ _ _ 6 4 3 3 3 2 4 3 .. -6 -5 -4 -3 _ -1 0 _ _ _ _ 5 3 3 3 2 4 3 4 .. 0 1 2 3 _ 5 6 _ _ _ _ 11 3 3 2 4 3 4 3 .. -1 0 1 2 _ 4 5 _ _ _ _ 10 3 2 4 3 4 3 3 .. -2 -1 0 1 _ 3 4 _ _ _ _ 9 2 3 3 3 3 4 4 .. 0 _ _ _ _ _ 6 _ 8 9 10 11 12 3 3 3 3 4 4 2 .. -8 _ _ _ _ _ -2 _ 0 1 2 3 4 3 3 3 4 4 2 3 .. -9 _ _ _ _ _ -3 _ -1 0 1 2 3 3 3 4 4 2 3 3 .. -10 _ _ _ _ _ -4 _ -2 -1 0 1 2 3 4 4 2 3 3 3 .. -11 _ _ _ _ _ -5 _ -3 -2 -1 0 1 4 4 2 3 3 3 3 .. -12 _ _ _ _ _ -6 _ -4 -3 -2 -1 0 4 2 3 3 3 3 4 .. -6 _ _ _ _ _ 0 _ 2 3 4 5 6 2 4 4 3 3 3 3 .. -4 -3 -2 -1 0 _ 2 _ _ _ _ _ 8 4 4 3 3 3 3 2 .. -12 -11 -10 -9 -8 _ -6 _ _ _ _ _ 0 4 3 3 3 3 2 4 .. -6 -5 -4 -3 -2 _ 0 _ _ _ _ _ 6 3 3 3 3 2 4 4 .. 0 1 2 3 4 _ 6 _ _ _ _ _ 12 3 3 3 2 4 4 3 .. -1 0 1 2 3 _ 5 _ _ _ _ _ 11 3 3 2 4 4 3 3 .. -2 -1 0 1 2 _ 4 _ _ _ _ _ 10 3 2 4 4 3 3 3 .. -3 -2 -1 0 1 _ 3 _ _ _ _ _ 9 2 3 3 3 3 3 5 .. 0 _ _ _ _ _ _ _ 8 9 10 11 12 13 3 3 3 3 3 5 2 .. -8 _ _ _ _ _ _ _ 0 1 2 3 4 5 3 3 3 3 5 2 3 .. -9 _ _ _ _ _ _ _ -1 0 1 2 3 4 3 3 3 5 2 3 3 .. -10 _ _ _ _ _ _ _ -2 -1 0 1 2 3 3 3 5 2 3 3 3 .. -11 _ _ _ _ _ _ _ -3 -2 -1 0 1 2 3 5 2 3 3 3 3 .. -12 _ _ _ _ _ _ _ -4 -3 -2 -1 0 1 5 2 3 3 3 3 3 .. -13 _ _ _ _ _ _ _ -5 -4 -3 -2 -1 0 2 5 3 3 3 3 3 .. -5 -4 -3 -2 -1 0 _ _ _ _ _ _ _ 8 5 3 3 3 3 3 2 .. -13 -12 -11 -10 -9 -8 _ _ _ _ _ _ _ 0 3 3 3 3 3 2 5 .. 0 1 2 3 4 5 _ _ _ _ _ _ _ 13 3 3 3 3 2 5 3 .. -1 0 1 2 3 4 _ _ _ _ _ _ _ 12 3 3 3 2 5 3 3 .. -2 -1 0 1 2 3 _ _ _ _ _ _ _ 11 3 3 2 5 3 3 3 .. -3 -2 -1 0 1 2 _ _ _ _ _ _ _ 10 3 2 5 3 3 3 3 .. -4 -3 -2 -1 0 1 _ _ _ _ _ _ _ 9 Did I miss any???
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<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 /> 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!<br /> <br /> <!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="Porcupine Temperament"></a><!-- ws:end:WikiTextHeadingRule:0 -->Porcupine Temperament</h1> <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 "tempered in" to 25/24, making it melodically useful instead of an "irritating mystery interval" which "introduces pitch drift".</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, "<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>"</li><li>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>."</li><li>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.)</li><li>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.)</li><li>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>"</li></ul><br /> <!-- ws:start:WikiTextHeadingRule:2:<h1> --><h1 id="toc1"><a name="Porcupine Chromaticism"></a><!-- ws:end:WikiTextHeadingRule:2 -->Porcupine Chromaticism</h1> <br /> 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 <a class="wiki_link" href="/140edo">140edo</a>, which is arguably an optimal tuning for Porcupine. Take a look:<br /> <br /> ...coming soon...<br /> <br /> <!-- ws:start:WikiTextHeadingRule:4:<h2> --><h2 id="toc2"><a name="Porcupine Chromaticism-Modes of Porcupine[7]"></a><!-- ws:end:WikiTextHeadingRule:4 -->Modes of Porcupine[7]</h2> <br /> The following modes are given in steps of 22edo. To generalize, 3 stands for the neutral third generating interval, and 4 stands for the wide major second. On the right is a contiguous chain of 7 tones separated by 6 iterations of the Porcupine generator.<br /> <br /> 3 3 3 3 3 3 4 .. 0 1 2 3 4 5 6<br /> 3 3 3 3 3 4 3 .. -1 0 1 2 3 4 5<br /> 3 3 3 3 4 3 3 .. -2 -1 0 1 2 3 4<br /> 3 3 3 4 3 3 3 .. -3 -2 -1 0 1 2 3<br /> 3 3 4 3 3 3 3 .. -4 -3 -2 -1 0 1 2<br /> 3 4 3 3 3 3 3 .. -5 -4 -3 -2 -1 0 1<br /> 4 3 3 3 3 3 3 .. -6 -5 -4 -3 -2 -1 0<br /> <br /> <!-- ws:start:WikiTextHeadingRule:6:<h2> --><h2 id="toc3"><a name="Porcupine Chromaticism-Chromatically-Altered Modes of Porcupine[7]"></a><!-- ws:end:WikiTextHeadingRule:6 -->Chromatically-Altered Modes of Porcupine[7]</h2> <br /> The following list includes all the modes that can be generated by shifting one tone of Porcupine[7] by one quartertone interval (chroma), which is one degree in 22edo.<br /> <br /> 2 4 3 3 3 3 4 .. 0 _ 2 3 4 5 6 _ 8<br /> 4 3 3 3 3 4 2 .. -8 _ -6 -5 -4 -3 -2 _ 0<br /> 3 3 3 3 4 2 4 .. -2 _ 0 1 2 3 4 _ 6<br /> 3 3 3 4 2 4 3 .. -3 _ -1 0 1 2 3 _ 5<br /> 3 3 4 2 4 3 3 .. -4 _ -2 -1 0 1 2 _ 4<br /> 3 4 2 4 3 3 3 .. -5 _ -3 -2 -1 0 1 _ 3<br /> 4 2 4 3 3 3 3 .. -6 _ -4 -3 -2 -1 0 _ 2<br /> <br /> 2 3 4 3 3 3 4 .. 0 _ _ 3 4 5 6 _ 8 9<br /> 3 4 3 3 3 4 2 .. -8 _ _ -5 -4 -3 -2 _ 0 1<br /> 4 3 3 3 4 2 3 .. -9 _ _ -6 -5 -4 -3 _ -1 0<br /> 3 3 3 4 2 3 4 .. -3 _ _ 0 1 2 3 _ 5 6<br /> 3 3 4 2 3 4 3 .. -4 _ _ -1 0 1 2 _ 4 5<br /> 3 4 2 3 4 3 3 .. -5 _ _ -2 -1 0 1 _ 3 4<br /> 4 2 3 4 3 3 3 .. -6 _ _ -3 -2 -1 0 _ 1 2<br /> <br /> 2 4 3 3 3 4 3 .. -1 0 _ 2 3 4 5 _ _ 8<br /> 4 3 3 3 4 3 2 .. -9 -8 _ -6 -5 -4 -3 _ _ 0<br /> 3 3 3 4 3 2 4 .. -3 -2 _ 0 1 2 3 _ _ 6<br /> 3 3 4 3 2 4 3 .. -4 -3 _ -1 0 1 2 _ _ 5<br /> 3 4 3 2 4 3 3 .. -5 -4 _ -2 -1 0 1 _ _ 4<br /> 4 3 2 4 3 3 3 .. -6 -5 _ -3 -2 -1 0 _ _ 3<br /> 3 2 4 3 3 3 4 .. 0 1 _ 3 4 5 6 _ _ 9<br /> <br /> 2 3 3 4 3 3 4 .. 0 _ _ _ 4 5 6 _ 8 9 10<br /> 3 3 4 3 3 4 2 .. -8 _ _ _ -4 -3 -2 _ 0 1 2<br /> 3 4 3 3 4 2 3 .. -9 _ _ _ -5 -4 -3 _ -1 0 1<br /> 4 3 3 4 2 3 3 .. -10 _ _ _ -6 -5 -4 _ -2 -1 0<br /> 3 3 4 2 3 3 4 .. -4 _ _ _ 0 1 2 _ 4 5 6<br /> 3 4 2 3 3 4 3 .. -5 _ _ _ -1 0 1 _ 3 4 5<br /> 4 2 3 3 4 3 3 .. -6 _ _ _ -2 -1 0 _ 2 3 4<br /> <br /> 2 4 3 3 4 3 3 .. -2 -1 0 _ 2 3 4 _ _ _ 8<br /> 4 3 3 4 3 3 2 .. -10 -9 -8 _ -6 -5 -4 _ _ _ 0<br /> 3 3 4 3 3 2 4 .. -4 -3 -2 _ 0 1 2 _ _ _ 6<br /> 3 4 3 3 2 4 3 .. -5 -4 -3 _ -1 0 1 _ _ _ 5<br /> 4 3 3 2 4 3 3 .. -6 -5 -4 _ -2 -1 0 _ _ _ 4<br /> 3 3 2 4 3 3 4 .. 0 1 2 _ 4 5 6 _ _ _ 10<br /> 3 2 4 3 3 4 3 .. -1 0 1 _ 3 4 5 _ _ _ 9<br /> <br /> 2 3 3 3 4 3 4 .. 0 _ _ _ _ 5 6 _ 8 9 10 11<br /> 3 3 3 4 3 4 2 .. -8 _ _ _ _ -3 -2 _ 0 1 2 3<br /> 3 3 4 3 4 2 3 .. -9 _ _ _ _ -4 -3 _ -1 0 1 2<br /> 3 4 3 4 2 3 3 .. -10 _ _ _ _ -5 -4 _ -2 -1 0 1<br /> 4 3 4 2 3 3 3 .. -11 _ _ _ _ -6 -5 _ -3 -2 -1 0<br /> 3 4 2 3 3 3 4 .. -5 _ _ _ _ 0 1 _ 3 4 5 6<br /> 4 2 3 3 3 4 3 .. -6 _ _ _ _ -1 0 _ 2 3 4 5<br /> <br /> 2 4 3 4 3 3 3 .. -3 -2 -1 0 _ 2 3 _ _ _ _ 8<br /> 4 3 4 3 3 3 2 .. -11 -10 -9 -8 _ -6 -5 _ _ _ _ 0<br /> 3 4 3 3 3 2 4 .. -5 -4 -3 -2 _ 0 1 _ _ _ _ 6<br /> 4 3 3 3 2 4 3 .. -6 -5 -4 -3 _ -1 0 _ _ _ _ 5<br /> 3 3 3 2 4 3 4 .. 0 1 2 3 _ 5 6 _ _ _ _ 11<br /> 3 3 2 4 3 4 3 .. -1 0 1 2 _ 4 5 _ _ _ _ 10<br /> 3 2 4 3 4 3 3 .. -2 -1 0 1 _ 3 4 _ _ _ _ 9<br /> <br /> 2 3 3 3 3 4 4 .. 0 _ _ _ _ _ 6 _ 8 9 10 11 12<br /> 3 3 3 3 4 4 2 .. -8 _ _ _ _ _ -2 _ 0 1 2 3 4<br /> 3 3 3 4 4 2 3 .. -9 _ _ _ _ _ -3 _ -1 0 1 2 3<br /> 3 3 4 4 2 3 3 .. -10 _ _ _ _ _ -4 _ -2 -1 0 1 2<br /> 3 4 4 2 3 3 3 .. -11 _ _ _ _ _ -5 _ -3 -2 -1 0 1<br /> 4 4 2 3 3 3 3 .. -12 _ _ _ _ _ -6 _ -4 -3 -2 -1 0<br /> 4 2 3 3 3 3 4 .. -6 _ _ _ _ _ 0 _ 2 3 4 5 6<br /> <br /> 2 4 4 3 3 3 3 .. -4 -3 -2 -1 0 _ 2 _ _ _ _ _ 8<br /> 4 4 3 3 3 3 2 .. -12 -11 -10 -9 -8 _ -6 _ _ _ _ _ 0<br /> 4 3 3 3 3 2 4 .. -6 -5 -4 -3 -2 _ 0 _ _ _ _ _ 6<br /> 3 3 3 3 2 4 4 .. 0 1 2 3 4 _ 6 _ _ _ _ _ 12<br /> 3 3 3 2 4 4 3 .. -1 0 1 2 3 _ 5 _ _ _ _ _ 11<br /> 3 3 2 4 4 3 3 .. -2 -1 0 1 2 _ 4 _ _ _ _ _ 10<br /> 3 2 4 4 3 3 3 .. -3 -2 -1 0 1 _ 3 _ _ _ _ _ 9<br /> <br /> 2 3 3 3 3 3 5 .. 0 _ _ _ _ _ _ _ 8 9 10 11 12 13<br /> 3 3 3 3 3 5 2 .. -8 _ _ _ _ _ _ _ 0 1 2 3 4 5<br /> 3 3 3 3 5 2 3 .. -9 _ _ _ _ _ _ _ -1 0 1 2 3 4<br /> 3 3 3 5 2 3 3 .. -10 _ _ _ _ _ _ _ -2 -1 0 1 2 3<br /> 3 3 5 2 3 3 3 .. -11 _ _ _ _ _ _ _ -3 -2 -1 0 1 2<br /> 3 5 2 3 3 3 3 .. -12 _ _ _ _ _ _ _ -4 -3 -2 -1 0 1<br /> 5 2 3 3 3 3 3 .. -13 _ _ _ _ _ _ _ -5 -4 -3 -2 -1 0<br /> <br /> 2 5 3 3 3 3 3 .. -5 -4 -3 -2 -1 0 _ _ _ _ _ _ _ 8<br /> 5 3 3 3 3 3 2 .. -13 -12 -11 -10 -9 -8 _ _ _ _ _ _ _ 0<br /> 3 3 3 3 3 2 5 .. 0 1 2 3 4 5 _ _ _ _ _ _ _ 13<br /> 3 3 3 3 2 5 3 .. -1 0 1 2 3 4 _ _ _ _ _ _ _ 12<br /> 3 3 3 2 5 3 3 .. -2 -1 0 1 2 3 _ _ _ _ _ _ _ 11<br /> 3 3 2 5 3 3 3 .. -3 -2 -1 0 1 2 _ _ _ _ _ _ _ 10<br /> 3 2 5 3 3 3 3 .. -4 -3 -2 -1 0 1 _ _ _ _ _ _ _ 9<br /> <br /> Did I miss any???</body></html>