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

==Modes of Porcupine[7] that have one chromatic alteration== 

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. Underscores represent gaps in the chain of Porcupine generators. Note that lowering a tone by one 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.

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 === Mike Battaglia's Porcupine[7] 0|6 #7
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 === Mike Battaglia's<span class="commentBody"> Porcupine[7] 3|3 #2</span>

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 === Mike Battaglia's Porcupine[7] 0|6 b4
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 === Mike Battaglia's Porcupine[7] 0|6 b7
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???

Original HTML content:

<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:&lt;h1&gt; --><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 &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:2:&lt;h1&gt; --><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:&lt;h2&gt; --><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:&lt;h2&gt; --><h2 id="toc3"><a name="Porcupine Chromaticism-Modes of Porcupine[7] that have one chromatic alteration"></a><!-- ws:end:WikiTextHeadingRule:6 -->Modes of Porcupine[7] that have one chromatic alteration</h2>
 <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. Underscores represent gaps in the chain of Porcupine generators. Note that lowering a tone by one 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.<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 === Mike Battaglia's Porcupine[7] 0|6 #7<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 === Mike Battaglia's<span class="commentBody"> Porcupine[7] 3|3 #2</span><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 === Mike Battaglia's Porcupine[7] 0|6 b4<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 === Mike Battaglia's Porcupine[7] 0|6 b7<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>