Andrew Heathwaite's MOS Investigations: Difference between revisions
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: The original revision id was <tt> | : The original revision id was <tt>287584396</tt>.<br> | ||
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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">This is a page for me, Andrew Heathwaite, to organize my thoughts and questions regarding [[MOSScales|Moment of Symmetry Scales]]. I'm using it primarily to provoke and organize conversations with myself. It's a sort of personal sandbox. If it provokes conversations with others, all the better! You *yes you* are more than welcome to correct obvious errors, add clearly-demarked related material, and comment through the comments tab -- ask questions, tell me where you think I'm totally bonkers, connect me to similar ideas that you may know about, give a hurrah or two -- whatever you find suitable. | <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">This is a page for me, Andrew Heathwaite, to organize my thoughts and questions regarding [[MOSScales|Moment of Symmetry Scales]]. I'm using it primarily to provoke and organize conversations with myself. It's a sort of personal sandbox. If it provokes conversations with others, all the better! You *yes you* are more than welcome to correct obvious errors, add clearly-demarked related material, and comment through the comments tab -- ask questions, tell me where you think I'm totally bonkers, connect me to similar ideas that you may know about, give a hurrah or two -- whatever you find suitable. | ||
==An Approach to Chromatic/Enharmonic MODMOS Scales== | |||
I am new to this and may be reinventing wheels. If you have anything to add or correct, please comment! | |||
[[MODMOS Scales]] generalize the class of scales which are not [[MOSScales|MOS]], but which have been obtained by applying a finite number of chromatic alterations to an MOS. In particular, the chromatic alterations are (usually) by "chroma," a small interval obtained by subtracting s from L. (c = L-s.) With strictly proper MOS Scales (which have 2s > L), the chroma is smaller than s (s > L-s). These are the scales I've been looking at recently. I have found that the MODMOS procedure produces interesting and useful scales in sensi[8] and miracle[10], as well as porcupine[7]. | |||
Mike Battaglia, whose enthusiastic promotion of the MODMOS scales of porcupine[7] has informed and inspired me, has proposed two new letters to stand for altered steps of MODMOS scales. "A" (short for "augmented") is a large step which has been widened by one chroma. (A = L+c.) Likewise, "d" (short for "diminished") is a small step which has been narrowed by one chroma. (d = s-c.) Thus, a MODMOS scale can be written like LsdLsss. I noticed that d and c can serve as "atoms" for deriving the other three steps, as follows: | |||
d = d | |||
s = d+c | |||
L = s+c = d+c+c = d+2c | |||
A = L+c = d+c+c+c = d+3c | |||
Notice that the chroma is not a step in its own right. This is particularly significant in these strictly-proper MOS scales, since the chroma is the smallest interval that is used. | |||
It may be useful to take subsets of larger MODMOS scales. In this case, it is likely that a 2-step interval will be taken as a single step in the subset scale. Of course, so could a 3-step interval and so on, but for now I'm looking at the 2-step intervals. Since the steps in our parent scale can be any of four varieties, we have the following possible 2-steps: | |||
dd = d+d = 2d | |||
sd = d+c+d = 2d+c | |||
ss = d+c+d+c = 2d+2c | |||
Ld = d+c+c+d = 2d+2c | |||
Ls = d+c+c+d+c = 2d+3c | |||
Ad = d+c+c+c+d = 2d+3c | |||
LL = d+c+c+d+c+c = 2d+4c | |||
As = d+c+c+c+d+c = 2d+4c | |||
AL = d+c+c+c+d+c+c = 2d+5c | |||
AA = d+c+c+c+d+c+c+c = 2d+6c | |||
Notice that ss=Ld, Ls=Ad, and LL=As. These equalities are always going to be true, as you can see by converting the steps into d's and c's. Other equalities may pop up, depending on the size of d and c -- we'll see that in our porcupine[7] examples below. | |||
Here is a chart showing all of the possible 1-steps and 2-steps in sensi[8] MODMOSes in [[46edo]] and miracle[10] MODMOSes in [[72edo]]. You can see that a lot more variety is available here than in a straight MOS scale. | |||
||~ step ||~ type ||~ # of d's ||~ # of c's ||~ degrees of sensi[8] ||~ cents value ||~ degrees of miracle[10] ||~ cents value || | |||
|| * || * || * || * || * || * || * || * || | |||
|| c || chroma || 0 || 1 || 2 || 52 || 2 || 33 || | |||
|| * || * || * || * || * || * || * || * || | |||
|| d || 1-step || 1 || 0 || 3 || 78 || 5 || 83 || | |||
|| s || 1-step || 1 || 1 || 5 || 130 || 7 || 117 || | |||
|| L || 1-step || 1 || 2 || 7 || 183 || 9 || 150 || | |||
|| A || 1-step || 1 || 3 || 9 || 235 || 11 || 183 || | |||
|| * || * || * || * || * || * || * || * || | |||
|| dd || 2-step || 2 || 0 || 6 || 157 || 10 || 167 || | |||
|| sd || 2-step || 2 || 1 || 8 || 209 || 12 || 200 || | |||
|| ss=Ld || 2-step || 2 || 2 || 10 || 261 || 14 || 233 || | |||
|| Ls=Ad || 2-step || 2 || 3 || 12 || 313 || 16 || 267 || | |||
|| LL=As || 2-step || 2 || 4 || 14 || 365 || 18 || 300 || | |||
|| AL || 2-step || 2 || 5 || 16 || 417 || 20 || 333 || | |||
|| AA || 2-step || 2 || 6 || 18 || 470 || 22 || 367 || | |||
In these examples, the largest 1-steps are larger than the smallest 2-steps, showing that a MODMOS scale of a proper MOS scale will not necessarily be proper. While a MODMOS scale may contain this variety of steps, it also may not. MODMOS scales sometimes produce permutations of the steps of the original MOS, in which case no diminished or augmented intervals appear at all. Thus, this chart is showing the maximum possible variety in 1-steps and 2-steps in sensi[8] and miracle[10] MODMOSes. | |||
In playing around with these scales, I found that the sensi[8] MODMOSes sound like albitonic (approximately diatonic-sized) scales -- 8 is not too far off from 7. I also found it pretty easy to distinguish the different types of steps. | |||
I found it more natural with miracle[10] MODMOSes, to take subsets. I also found it more difficult to distinguish between steps, as they are closer together. Gene Smith's "Muddle" scales make sense to apply here. That means taking a MOS-like subset of a non-equal scale. For example, take the Miracle[10] MODMOS ssdLsssLss. We may group the steps (ss)(d)(Ls)(s)(sL)(s)(s). This mirrors the structure of the MOS scale of [[10edo]] that goes 2 1 2 1 2 1 1. Since there are 7 different rotations of this MOS pattern, we may apply it in 7 different ways in our MODMOS. Here it is in steps of 72edo: | |||
parent scale: **7 7 5 9 7 7 7 9 7 7** | |||
children: | |||
form 2 1 2 1 2 1 1: **14 5 16 7 16 7 7** | |||
form 1 2 1 2 1 1 2: **7 12 9 14 7 9 14** | |||
form 2 1 2 1 1 2 1: **14 5 16 7 7 16 7** | |||
form 1 2 1 1 2 1 2: **7 12 9 7 14 9 14** | |||
form 2 1 1 2 1 2 1: **14 5 9 14 7 16 7** | |||
form 1 1 2 1 2 1 2: **7 7 14 7 14 9 14** | |||
It also works to take every other note, generating two different pentatonics with no tones in common. | |||
So I did this same analysis with porcupine[7], and I used three different [[edo]]s: [[37edo]], [[22edo]] and [[15edo]]: | |||
||~ step ||~ degrees of porcupine[7] in 37edo ||~ cents value ||~ degrees of porcupine[7] in 22edo ||~ cents value ||~ degrees of porcupine[7] in 15edo ||~ cents value || | |||
|| * || * || * || * || * || * || * || | |||
|| c || 2 || 65 || 1 || 55 || 1 || 80 || | |||
|| * || * || * || * || * || * || * || | |||
|| d || 3 || 97 || 2 || 109 || 1 || 80 || | |||
|| s || 5 || 162 || 3 || 164 || 2 || 160 || | |||
|| L || 7 || 227 || 4 || 218 || 3 || 240 || | |||
|| A || 9 || 292 || 5 || 273 || 4 || 320 || | |||
|| * || * || * || * || * || * || * || | |||
|| dd || 6 || 195 || 4 || 218 || 2 || 160 || | |||
|| sd || 8 || 259 || 5 || 273 || 3 || 240 || | |||
|| ss=Ld || 10 || 324 || 6 || 327 || 4 || 320 || | |||
|| Ls=Ad || 12 || 389 || 7 || 382 || 5 || 400 || | |||
|| LL=As || 14 || 454 || 8 || 436 || 6 || 480 || | |||
|| AL || 16 || 519 || 9 || 491 || 7 || 560 || | |||
|| AA || 18 || 584 || 10 || 545 || 8 || 640 || | |||
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. | |||
==Expanding on "Maximal Evenness"== | ==Expanding on "Maximal Evenness"== | ||
| Line 285: | Line 373: | ||
<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;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Andrew Heathwaite's MOS Investigations</title></head><body>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>. I'm using it primarily to provoke and organize conversations with myself. It's a sort of personal sandbox. If it provokes conversations with others, all the better! You *yes you* are more than welcome to correct obvious errors, add clearly-demarked related material, and comment through the comments tab -- ask questions, tell me where you think I'm totally bonkers, connect me to similar ideas that you may know about, give a hurrah or two -- whatever you find suitable.<br /> | <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;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Andrew Heathwaite's MOS Investigations</title></head><body>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>. I'm using it primarily to provoke and organize conversations with myself. It's a sort of personal sandbox. If it provokes conversations with others, all the better! You *yes you* are more than welcome to correct obvious errors, add clearly-demarked related material, and comment through the comments tab -- ask questions, tell me where you think I'm totally bonkers, connect me to similar ideas that you may know about, give a hurrah or two -- whatever you find suitable.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-Expanding on &quot;Maximal Evenness&quot;"></a><!-- ws:end:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-An Approach to Chromatic/Enharmonic MODMOS Scales"></a><!-- ws:end:WikiTextHeadingRule:0 -->An Approach to Chromatic/Enharmonic MODMOS Scales</h2> | ||
<br /> | |||
I am new to this and may be reinventing wheels. If you have anything to add or correct, please comment!<br /> | |||
<br /> | |||
<a class="wiki_link" href="/MODMOS%20Scales">MODMOS Scales</a> generalize the class of scales which are not <a class="wiki_link" href="/MOSScales">MOS</a>, but which have been obtained by applying a finite number of chromatic alterations to an MOS. In particular, the chromatic alterations are (usually) by &quot;chroma,&quot; a small interval obtained by subtracting s from L. (c = L-s.) With strictly proper MOS Scales (which have 2s &gt; L), the chroma is smaller than s (s &gt; L-s). These are the scales I've been looking at recently. I have found that the MODMOS procedure produces interesting and useful scales in sensi[8] and miracle[10], as well as porcupine[7].<br /> | |||
<br /> | |||
Mike Battaglia, whose enthusiastic promotion of the MODMOS scales of porcupine[7] has informed and inspired me, has proposed two new letters to stand for altered steps of MODMOS scales. &quot;A&quot; (short for &quot;augmented&quot;) is a large step which has been widened by one chroma. (A = L+c.) Likewise, &quot;d&quot; (short for &quot;diminished&quot;) is a small step which has been narrowed by one chroma. (d = s-c.) Thus, a MODMOS scale can be written like LsdLsss. I noticed that d and c can serve as &quot;atoms&quot; for deriving the other three steps, as follows:<br /> | |||
<br /> | |||
d = d<br /> | |||
s = d+c<br /> | |||
L = s+c = d+c+c = d+2c<br /> | |||
A = L+c = d+c+c+c = d+3c<br /> | |||
<br /> | |||
Notice that the chroma is not a step in its own right. This is particularly significant in these strictly-proper MOS scales, since the chroma is the smallest interval that is used.<br /> | |||
<br /> | |||
It may be useful to take subsets of larger MODMOS scales. In this case, it is likely that a 2-step interval will be taken as a single step in the subset scale. Of course, so could a 3-step interval and so on, but for now I'm looking at the 2-step intervals. Since the steps in our parent scale can be any of four varieties, we have the following possible 2-steps:<br /> | |||
<br /> | |||
dd = d+d = 2d<br /> | |||
sd = d+c+d = 2d+c<br /> | |||
ss = d+c+d+c = 2d+2c<br /> | |||
Ld = d+c+c+d = 2d+2c<br /> | |||
Ls = d+c+c+d+c = 2d+3c<br /> | |||
Ad = d+c+c+c+d = 2d+3c<br /> | |||
LL = d+c+c+d+c+c = 2d+4c<br /> | |||
As = d+c+c+c+d+c = 2d+4c<br /> | |||
AL = d+c+c+c+d+c+c = 2d+5c<br /> | |||
AA = d+c+c+c+d+c+c+c = 2d+6c<br /> | |||
<br /> | |||
Notice that ss=Ld, Ls=Ad, and LL=As. These equalities are always going to be true, as you can see by converting the steps into d's and c's. Other equalities may pop up, depending on the size of d and c -- we'll see that in our porcupine[7] examples below.<br /> | |||
<br /> | |||
Here is a chart showing all of the possible 1-steps and 2-steps in sensi[8] MODMOSes in <a class="wiki_link" href="/46edo">46edo</a> and miracle[10] MODMOSes in <a class="wiki_link" href="/72edo">72edo</a>. You can see that a lot more variety is available here than in a straight MOS scale.<br /> | |||
<br /> | |||
<table class="wiki_table"> | |||
<tr> | |||
<th>step<br /> | |||
</th> | |||
<th>type<br /> | |||
</th> | |||
<th><ol><li>of d's</li></ol></th> | |||
<th><ol><li>of c's</li></ol></th> | |||
<th>degrees of sensi[8]<br /> | |||
</th> | |||
<th>cents value<br /> | |||
</th> | |||
<th>degrees of miracle[10]<br /> | |||
</th> | |||
<th>cents value<br /> | |||
</th> | |||
</tr> | |||
<tr> | |||
<td>*<br /> | |||
</td> | |||
<td>*<br /> | |||
</td> | |||
<td>*<br /> | |||
</td> | |||
<td>*<br /> | |||
</td> | |||
<td>*<br /> | |||
</td> | |||
<td>*<br /> | |||
</td> | |||
<td>*<br /> | |||
</td> | |||
<td>*<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>c<br /> | |||
</td> | |||
<td>chroma<br /> | |||
</td> | |||
<td>0<br /> | |||
</td> | |||
<td>1<br /> | |||
</td> | |||
<td>2<br /> | |||
</td> | |||
<td>52<br /> | |||
</td> | |||
<td>2<br /> | |||
</td> | |||
<td>33<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>*<br /> | |||
</td> | |||
<td>*<br /> | |||
</td> | |||
<td>*<br /> | |||
</td> | |||
<td>*<br /> | |||
</td> | |||
<td>*<br /> | |||
</td> | |||
<td>*<br /> | |||
</td> | |||
<td>*<br /> | |||
</td> | |||
<td>*<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>d<br /> | |||
</td> | |||
<td>1-step<br /> | |||
</td> | |||
<td>1<br /> | |||
</td> | |||
<td>0<br /> | |||
</td> | |||
<td>3<br /> | |||
</td> | |||
<td>78<br /> | |||
</td> | |||
<td>5<br /> | |||
</td> | |||
<td>83<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>s<br /> | |||
</td> | |||
<td>1-step<br /> | |||
</td> | |||
<td>1<br /> | |||
</td> | |||
<td>1<br /> | |||
</td> | |||
<td>5<br /> | |||
</td> | |||
<td>130<br /> | |||
</td> | |||
<td>7<br /> | |||
</td> | |||
<td>117<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>L<br /> | |||
</td> | |||
<td>1-step<br /> | |||
</td> | |||
<td>1<br /> | |||
</td> | |||
<td>2<br /> | |||
</td> | |||
<td>7<br /> | |||
</td> | |||
<td>183<br /> | |||
</td> | |||
<td>9<br /> | |||
</td> | |||
<td>150<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>A<br /> | |||
</td> | |||
<td>1-step<br /> | |||
</td> | |||
<td>1<br /> | |||
</td> | |||
<td>3<br /> | |||
</td> | |||
<td>9<br /> | |||
</td> | |||
<td>235<br /> | |||
</td> | |||
<td>11<br /> | |||
</td> | |||
<td>183<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>*<br /> | |||
</td> | |||
<td>*<br /> | |||
</td> | |||
<td>*<br /> | |||
</td> | |||
<td>*<br /> | |||
</td> | |||
<td>*<br /> | |||
</td> | |||
<td>*<br /> | |||
</td> | |||
<td>*<br /> | |||
</td> | |||
<td>*<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>dd<br /> | |||
</td> | |||
<td>2-step<br /> | |||
</td> | |||
<td>2<br /> | |||
</td> | |||
<td>0<br /> | |||
</td> | |||
<td>6<br /> | |||
</td> | |||
<td>157<br /> | |||
</td> | |||
<td>10<br /> | |||
</td> | |||
<td>167<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>sd<br /> | |||
</td> | |||
<td>2-step<br /> | |||
</td> | |||
<td>2<br /> | |||
</td> | |||
<td>1<br /> | |||
</td> | |||
<td>8<br /> | |||
</td> | |||
<td>209<br /> | |||
</td> | |||
<td>12<br /> | |||
</td> | |||
<td>200<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>ss=Ld<br /> | |||
</td> | |||
<td>2-step<br /> | |||
</td> | |||
<td>2<br /> | |||
</td> | |||
<td>2<br /> | |||
</td> | |||
<td>10<br /> | |||
</td> | |||
<td>261<br /> | |||
</td> | |||
<td>14<br /> | |||
</td> | |||
<td>233<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>Ls=Ad<br /> | |||
</td> | |||
<td>2-step<br /> | |||
</td> | |||
<td>2<br /> | |||
</td> | |||
<td>3<br /> | |||
</td> | |||
<td>12<br /> | |||
</td> | |||
<td>313<br /> | |||
</td> | |||
<td>16<br /> | |||
</td> | |||
<td>267<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>LL=As<br /> | |||
</td> | |||
<td>2-step<br /> | |||
</td> | |||
<td>2<br /> | |||
</td> | |||
<td>4<br /> | |||
</td> | |||
<td>14<br /> | |||
</td> | |||
<td>365<br /> | |||
</td> | |||
<td>18<br /> | |||
</td> | |||
<td>300<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>AL<br /> | |||
</td> | |||
<td>2-step<br /> | |||
</td> | |||
<td>2<br /> | |||
</td> | |||
<td>5<br /> | |||
</td> | |||
<td>16<br /> | |||
</td> | |||
<td>417<br /> | |||
</td> | |||
<td>20<br /> | |||
</td> | |||
<td>333<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>AA<br /> | |||
</td> | |||
<td>2-step<br /> | |||
</td> | |||
<td>2<br /> | |||
</td> | |||
<td>6<br /> | |||
</td> | |||
<td>18<br /> | |||
</td> | |||
<td>470<br /> | |||
</td> | |||
<td>22<br /> | |||
</td> | |||
<td>367<br /> | |||
</td> | |||
</tr> | |||
</table> | |||
<br /> | |||
In these examples, the largest 1-steps are larger than the smallest 2-steps, showing that a MODMOS scale of a proper MOS scale will not necessarily be proper. While a MODMOS scale may contain this variety of steps, it also may not. MODMOS scales sometimes produce permutations of the steps of the original MOS, in which case no diminished or augmented intervals appear at all. Thus, this chart is showing the maximum possible variety in 1-steps and 2-steps in sensi[8] and miracle[10] MODMOSes.<br /> | |||
<br /> | |||
In playing around with these scales, I found that the sensi[8] MODMOSes sound like albitonic (approximately diatonic-sized) scales -- 8 is not too far off from 7. I also found it pretty easy to distinguish the different types of steps.<br /> | |||
<br /> | |||
I found it more natural with miracle[10] MODMOSes, to take subsets. I also found it more difficult to distinguish between steps, as they are closer together. Gene Smith's &quot;Muddle&quot; scales make sense to apply here. That means taking a MOS-like subset of a non-equal scale. For example, take the Miracle[10] MODMOS ssdLsssLss. We may group the steps (ss)(d)(Ls)(s)(sL)(s)(s). This mirrors the structure of the MOS scale of <a class="wiki_link" href="/10edo">10edo</a> that goes 2 1 2 1 2 1 1. Since there are 7 different rotations of this MOS pattern, we may apply it in 7 different ways in our MODMOS. Here it is in steps of 72edo:<br /> | |||
<br /> | |||
parent scale: <strong>7 7 5 9 7 7 7 9 7 7</strong><br /> | |||
<br /> | |||
children:<br /> | |||
form 2 1 2 1 2 1 1: <strong>14 5 16 7 16 7 7</strong><br /> | |||
form 1 2 1 2 1 1 2: <strong>7 12 9 14 7 9 14</strong><br /> | |||
form 2 1 2 1 1 2 1: <strong>14 5 16 7 7 16 7</strong><br /> | |||
form 1 2 1 1 2 1 2: <strong>7 12 9 7 14 9 14</strong><br /> | |||
form 2 1 1 2 1 2 1: <strong>14 5 9 14 7 16 7</strong><br /> | |||
form 1 1 2 1 2 1 2: <strong>7 7 14 7 14 9 14</strong><br /> | |||
<br /> | |||
It also works to take every other note, generating two different pentatonics with no tones in common.<br /> | |||
<br /> | |||
So I did this same analysis with porcupine[7], and I used three different <a class="wiki_link" href="/edo">edo</a>s: <a class="wiki_link" href="/37edo">37edo</a>, <a class="wiki_link" href="/22edo">22edo</a> and <a class="wiki_link" href="/15edo">15edo</a>:<br /> | |||
<br /> | |||
<table class="wiki_table"> | |||
<tr> | |||
<th>step<br /> | |||
</th> | |||
<th>degrees of porcupine[7] in 37edo<br /> | |||
</th> | |||
<th>cents value<br /> | |||
</th> | |||
<th>degrees of porcupine[7] in 22edo<br /> | |||
</th> | |||
<th>cents value<br /> | |||
</th> | |||
<th>degrees of porcupine[7] in 15edo<br /> | |||
</th> | |||
<th>cents value<br /> | |||
</th> | |||
</tr> | |||
<tr> | |||
<td>*<br /> | |||
</td> | |||
<td>*<br /> | |||
</td> | |||
<td>*<br /> | |||
</td> | |||
<td>*<br /> | |||
</td> | |||
<td>*<br /> | |||
</td> | |||
<td>*<br /> | |||
</td> | |||
<td>*<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>c<br /> | |||
</td> | |||
<td>2<br /> | |||
</td> | |||
<td>65<br /> | |||
</td> | |||
<td>1<br /> | |||
</td> | |||
<td>55<br /> | |||
</td> | |||
<td>1<br /> | |||
</td> | |||
<td>80<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>*<br /> | |||
</td> | |||
<td>*<br /> | |||
</td> | |||
<td>*<br /> | |||
</td> | |||
<td>*<br /> | |||
</td> | |||
<td>*<br /> | |||
</td> | |||
<td>*<br /> | |||
</td> | |||
<td>*<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>d<br /> | |||
</td> | |||
<td>3<br /> | |||
</td> | |||
<td>97<br /> | |||
</td> | |||
<td>2<br /> | |||
</td> | |||
<td>109<br /> | |||
</td> | |||
<td>1<br /> | |||
</td> | |||
<td>80<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>s<br /> | |||
</td> | |||
<td>5<br /> | |||
</td> | |||
<td>162<br /> | |||
</td> | |||
<td>3<br /> | |||
</td> | |||
<td>164<br /> | |||
</td> | |||
<td>2<br /> | |||
</td> | |||
<td>160<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>L<br /> | |||
</td> | |||
<td>7<br /> | |||
</td> | |||
<td>227<br /> | |||
</td> | |||
<td>4<br /> | |||
</td> | |||
<td>218<br /> | |||
</td> | |||
<td>3<br /> | |||
</td> | |||
<td>240<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>A<br /> | |||
</td> | |||
<td>9<br /> | |||
</td> | |||
<td>292<br /> | |||
</td> | |||
<td>5<br /> | |||
</td> | |||
<td>273<br /> | |||
</td> | |||
<td>4<br /> | |||
</td> | |||
<td>320<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>*<br /> | |||
</td> | |||
<td>*<br /> | |||
</td> | |||
<td>*<br /> | |||
</td> | |||
<td>*<br /> | |||
</td> | |||
<td>*<br /> | |||
</td> | |||
<td>*<br /> | |||
</td> | |||
<td>*<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>dd<br /> | |||
</td> | |||
<td>6<br /> | |||
</td> | |||
<td>195<br /> | |||
</td> | |||
<td>4<br /> | |||
</td> | |||
<td>218<br /> | |||
</td> | |||
<td>2<br /> | |||
</td> | |||
<td>160<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>sd<br /> | |||
</td> | |||
<td>8<br /> | |||
</td> | |||
<td>259<br /> | |||
</td> | |||
<td>5<br /> | |||
</td> | |||
<td>273<br /> | |||
</td> | |||
<td>3<br /> | |||
</td> | |||
<td>240<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>ss=Ld<br /> | |||
</td> | |||
<td>10<br /> | |||
</td> | |||
<td>324<br /> | |||
</td> | |||
<td>6<br /> | |||
</td> | |||
<td>327<br /> | |||
</td> | |||
<td>4<br /> | |||
</td> | |||
<td>320<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>Ls=Ad<br /> | |||
</td> | |||
<td>12<br /> | |||
</td> | |||
<td>389<br /> | |||
</td> | |||
<td>7<br /> | |||
</td> | |||
<td>382<br /> | |||
</td> | |||
<td>5<br /> | |||
</td> | |||
<td>400<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>LL=As<br /> | |||
</td> | |||
<td>14<br /> | |||
</td> | |||
<td>454<br /> | |||
</td> | |||
<td>8<br /> | |||
</td> | |||
<td>436<br /> | |||
</td> | |||
<td>6<br /> | |||
</td> | |||
<td>480<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>AL<br /> | |||
</td> | |||
<td>16<br /> | |||
</td> | |||
<td>519<br /> | |||
</td> | |||
<td>9<br /> | |||
</td> | |||
<td>491<br /> | |||
</td> | |||
<td>7<br /> | |||
</td> | |||
<td>560<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>AA<br /> | |||
</td> | |||
<td>18<br /> | |||
</td> | |||
<td>584<br /> | |||
</td> | |||
<td>10<br /> | |||
</td> | |||
<td>545<br /> | |||
</td> | |||
<td>8<br /> | |||
</td> | |||
<td>640<br /> | |||
</td> | |||
</tr> | |||
</table> | |||
<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 /> | |||
<!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="x-Expanding on &quot;Maximal Evenness&quot;"></a><!-- ws:end:WikiTextHeadingRule:2 -->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 /> | ||
| Line 346: | Line 1,042: | ||
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:4:&lt;h2&gt; --><h2 id="toc2"><a name="x-MOS Scales with similar generators"></a><!-- ws:end:WikiTextHeadingRule:4 -->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:876:&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:876 --><br /> | ||
<!-- ws:start:WikiTextLocalImageRule: | <!-- ws:start:WikiTextLocalImageRule:877:&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:877 --><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:6:&lt;h2&gt; --><h2 id="toc3"><a name="x-Notes on Keenan Pepper's Diatonic-like MOS Scales"></a><!-- ws:end:WikiTextHeadingRule:6 -->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 /> | ||
| Line 623: | Line 1,319: | ||
<br /> | <br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc4"><a name="Porcupine Temperament"></a><!-- ws:end:WikiTextHeadingRule:8 -->Porcupine Temperament</h1> | ||
<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>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:10:&lt;h1&gt; --><h1 id="toc5"><a name="Porcupine Chromaticism"></a><!-- ws:end:WikiTextHeadingRule:10 -->Porcupine Chromaticism</h1> | ||
<br /> | <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 /> | 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: | <!-- ws:start:WikiTextLocalImageRule:878:&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:878 --><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 /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:12:&lt;h2&gt; --><h2 id="toc6"><a name="Porcupine Chromaticism-Modes of Porcupine[7]"></a><!-- ws:end:WikiTextHeadingRule:12 -->Modes of Porcupine[7]</h2> | ||
<br /> | <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 /> | 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 /> | ||
| Line 647: | Line 1,343: | ||
<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 /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:14:&lt;h2&gt; --><h2 id="toc7"><a name="Porcupine Chromaticism-Modes of Porcupine[7] that have one chromatic alteration"></a><!-- ws:end:WikiTextHeadingRule:14 -->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 /> | ||
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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:16:&lt;h1&gt; --><h1 id="toc8"><a name="Orwell[9], meet Porcupine[7]"></a><!-- ws:end:WikiTextHeadingRule:16 -->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:18:&lt;h1&gt; --><h1 id="toc9"><a name="Names for steps"></a><!-- ws:end:WikiTextHeadingRule:18 -->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 /> | ||