Andrew Heathwaite's MOS Investigations

IMPORTED REVISION FROM WIKISPACES

This is an imported revision from Wikispaces. The revision metadata is included below for reference:

This revision was by author Andrew_Heathwaite and made on 2011-11-01 13:08:37 UTC.

The original revision id was 270703714.

The revision comment was:

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.

Original Wikitext content:

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...

Hm, try as I might, I can't get wikispaces to upload this image. Nevermind then.

==Modes of Porcupine[7]== 

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 "-3" in the chain has a perfect fifth over the bass.

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. 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.

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 === one of Andrew's faves**
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 === one of Andrew's faves**
**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**

Update: Mike Battaglia has made a dedicated page for explaining these modes -- yay! -- see [[Porcupine Temperament Modal Harmony]].

=Orwell[9], meet Porcupine[7]= 

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 "chroma," and it's generated by taking L-s: so in 22edo we have 3\22-2\22=1\22. We wind up with either:
# A permutation of the four large and five small steps, eg. 3 3 2 2 3 2 3 2 2
## How many of these are there? Does anyone know a formula for finding the number of possible permutations when some of the items are interchangable? Here the question is, how many permutations can we make of 4 items of Type A and 5 items of Type B?
# A scale with three step sizes: large, small, and smaller, eg. 3 2 3 2 3 2 3 3 1
## In 22edo, our "smaller" step is the same as our "chroma" (which is the interval that we alter a tone by to produce a MODMOS, L-s). However, this is not the case in larger edos! Look at [[31edo]], where our initial scale is 4 3 4 3 4 3 4 3 3. Now our chroma is 4\31-3\31=1\31 and our "smaller" step is 2\31: 4 3 4 3 4 3 4 4 2! We get our "smaller" step by starting with s (3\31) and taking away a chroma (1\31), so we have 3\31-1\31=2\31.
## So what should we call the "smaller" interval in our scale? Maybe some kind of diminished something-or-another?
# A scale with four steps sizes: large, small, larger and smaller, eg. 4 1 3 2 3 2 3 2 2
## This is generated by starting with L and adding a chroma, so in 22edo it's 3\22+1\22=4\22. In 31edo, that would be 4\31+1\31=5\31, and the scale in question would be 5 2 4 3 4 3 4 3 3.
## So what should we call the "larger" step? Some kind of augmented something-or-another?
## Note that in 22edo, our "larger" step, 4\22, is the same as two of our small steps (2\22+2\22=4\22), even though we generated our "larger" step by adding a chroma to a large step (3\22+1\22). In 31edo, our "larger" step is NOT the same as two of our small steps (4\31+1\31=5\31 does not equal 3\31+3\31=6\31)!

So we can take advantage of the fact that two small steps in 22edo's Orwell[9] (2\22) make one "larger" step (4\22). If 9 tones is a few too many, we can turn some 2+2's into 4's. So for instance, the first example above goes:

3 3 2 2 3 2 3 2 2
3 3 4 3 2 3 4.

But check it out! 3 3 4 3 2 3 4 is a MODMOS of Porcupine[7]! Here's how we can get it by chromatically-altering Porcupine[7] one tone at a time:

3 3 3 3 3 3 4
3 3 4 2 3 3 4
3 3 4 3 2 3 4

And we see, not surprisingly, that this doesn't work the same way in 31edo.

Start with a MODMOS of Orwell[9]: 4 4 3 3 4 3 4 3 3
Combine small steps: 4 4 6 4 3 4 6

4 4 4 4 4 4 7 is as close as we can get to Porcupine[7], and it sure ain't the same. Our chroma (L-s) is 3\31, really different!
4 4 7 1 4 4 7
4 4 7 1 4 1 7

Not even close!

=Names for steps= 

This is getting silly! We need better names.....

So, as proposed on the page for [[MODMOS Scales]], we could call L-s a "chroma" and abbreviate it "c". That's a good start.

We have another step that's s-c, or s-(L-s) = s-L+s = 2s-L. In Porcupine[7] in 22edo, that's:
s-c = 3\22-1\22 = 2\22
or
2s-L = 2(3\22)-4\22 = 6\22-4\22 = 2\22.
Some kind of diminished step?

And we have another step that's L+c, or L+(L-s) = 2L-s. In Porcupine[7] in 22edo, that's:
L+c = 4\22+1\22 = 5\22
or
2L-s = 2(4\22)-3\22 = 8\22-3\22 = 5\22.
Some kind of augmented step?

Mike Battaglia proposes (at least in the case of Porcupine) "d" for s-c and "A" for L+c....

<span class="commentBody">So I posted to XA: "Ok, thinking it over, A and d are interesting choices to describe the additional steps, since they're so general. "d" is the small step minus one chroma, or s-c. But since the chroma itself is L-s, we can define d directly in terms of L a</span><span class="text_exposed_show">nd s as 2s-L. Meanwhile, "A" is the large step plus one chroma, or L+c. Described in terms of L and s, "A" is 2L-s. This allows us to quickly compute c, A, and d quickly, given L and s. As one example, Orwell[9] in 53edo has L=7 and s=5. So c=L-s=7-5=2; d=2s-L=2(5)-7=3; and A=2L-s=2(7)-5=9. Sure enough, a MODMOS of Orwell[9] with all four of these steps can be easily generated. Start with 7 5 7 5 7 5 7 5 5 and shift the second tone up by one chroma (which we computed to be 2\53), producing 9 3 7 5 7 5 7 5 5, which generalizes to AdLsLsLss! (Note that in 53edo Orwell[9] 2s=10 and A=9, not equal; while in 22edo's version of Orwell[9] 2s=4 and A=4, a potential 22edo Orwell pun.)</span><span class="commentBody"> ... ‎(Oh, and another pun is possible here, since c and d are both 1 degree in 22edo but 2 and 3 degrees, respectively, in 53edo.)"</span>

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 />
Hm, try as I might, I can't get wikispaces to upload this image. Nevermind then.<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. 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 />
<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 />
<strong>3 3 3 4 3 3 3 .. -3 -2 -1 0 1 2 3</strong><br />
<strong>3 3 4 3 3 3 3 .. -4 -3 -2 -1 0 1 2</strong><br />
<strong>3 4 3 3 3 3 3 .. -5 -4 -3 -2 -1 0 1</strong><br />
<strong>4 3 3 3 3 3 3 .. -6 -5 -4 -3 -2 -1 0</strong><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. 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 />
<br />
2 4 3 3 3 3 4 .. 0 _ 2 3 4 5 6 _ 8<br />
<strong>4 3 3 3 3 4 2 .. -8 _ -6 -5 -4 -3 -2 _ 0 === Mike Battaglia's Porcupine[7] 0|6 #7</strong><br />
3 3 3 3 4 2 4 .. -2 _ 0 1 2 3 4 _ 6<br />
<strong>3 3 3 4 2 4 3 .. -3 _ -1 0 1 2 3 _ 5</strong><br />
3 3 4 2 4 3 3 .. -4 _ -2 -1 0 1 2 _ 4<br />
<strong>3 4 2 4 3 3 3 .. -5 _ -3 -2 -1 0 1 _ 3</strong><br />
<strong>4 2 4 3 3 3 3 .. -6 _ -4 -3 -2 -1 0 _ 2</strong><br />
<br />
2 3 4 3 3 3 4 .. 0 _ _ 3 4 5 6 _ 8 9<br />
<strong>3 4 3 3 3 4 2 .. -8 _ _ -5 -4 -3 -2 _ 0 1</strong><br />
<strong>4 3 3 3 4 2 3 .. -9 _ _ -6 -5 -4 -3 _ -1 0</strong><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 />
<strong>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></strong><br />
<br />
2 4 3 3 3 4 3 .. -1 0 _ 2 3 4 5 _ _ 8<br />
<strong>4 3 3 3 4 3 2 .. -9 -8 _ -6 -5 -4 -3 _ _ 0</strong><br />
<strong>3 3 3 4 3 2 4 .. -3 -2 _ 0 1 2 3 _ _ 6</strong><br />
<strong>3 3 4 3 2 4 3 .. -4 -3 _ -1 0 1 2 _ _ 5 === one of Andrew's faves</strong><br />
3 4 3 2 4 3 3 .. -5 -4 _ -2 -1 0 1 _ _ 4<br />
<strong>4 3 2 4 3 3 3 .. -6 -5 _ -3 -2 -1 0 _ _ 3 === Mike Battaglia's Porcupine[7] 0|6 b4</strong><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 />
<strong>3 3 4 3 3 4 2 .. -8 _ _ _ -4 -3 -2 _ 0 1 2</strong><br />
<strong>3 4 3 3 4 2 3 .. -9 _ _ _ -5 -4 -3 _ -1 0 1</strong><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 />
<strong>3 3 4 3 3 2 4 .. -4 -3 -2 _ 0 1 2 _ _ _ 6</strong><br />
<strong>3 4 3 3 2 4 3 .. -5 -4 -3 _ -1 0 1 _ _ _ 5</strong><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 />
<strong>3 3 3 4 3 4 2 .. -8 _ _ _ _ -3 -2 _ 0 1 2 3</strong><br />
<strong>3 3 4 3 4 2 3 .. -9 _ _ _ _ -4 -3 _ -1 0 1 2</strong><br />
3 4 3 4 2 3 3 .. -10 _ _ _ _ -5 -4 _ -2 -1 0 1<br />
<strong>4 3 4 2 3 3 3 .. -11 _ _ _ _ -6 -5 _ -3 -2 -1 0</strong><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 />
<strong>2 4 3 4 3 3 3 .. -3 -2 -1 0 _ 2 3 _ _ _ _ 8</strong><br />
4 3 4 3 3 3 2 .. -11 -10 -9 -8 _ -6 -5 _ _ _ _ 0<br />
<strong>3 4 3 3 3 2 4 .. -5 -4 -3 -2 _ 0 1 _ _ _ _ 6 === one of Andrew's faves</strong><br />
<strong>4 3 3 3 2 4 3 .. -6 -5 -4 -3 _ -1 0 _ _ _ _ 5</strong><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 />
<strong>3 3 3 4 4 2 3 .. -9 _ _ _ _ _ -3 _ -1 0 1 2 3</strong><br />
3 3 4 4 2 3 3 .. -10 _ _ _ _ _ -4 _ -2 -1 0 1 2<br />
<strong>3 4 4 2 3 3 3 .. -11 _ _ _ _ _ -5 _ -3 -2 -1 0 1</strong><br />
<strong>4 4 2 3 3 3 3 .. -12 _ _ _ _ _ -6 _ -4 -3 -2 -1 0</strong><br />
4 2 3 3 3 3 4 .. -6 _ _ _ _ _ 0 _ 2 3 4 5 6<br />
<br />
<strong>2 4 4 3 3 3 3 .. -4 -3 -2 -1 0 _ 2 _ _ _ _ _ 8</strong><br />
4 4 3 3 3 3 2 .. -12 -11 -10 -9 -8 _ -6 _ _ _ _ _ 0<br />
<strong>4 3 3 3 3 2 4 .. -6 -5 -4 -3 -2 _ 0 _ _ _ _ _ 6 === Mike Battaglia's Porcupine[7] 0|6 b7</strong><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 />
<strong>3 2 4 4 3 3 3 .. -3 -2 -1 0 1 _ 3 _ _ _ _ _ 9</strong><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 />
<strong>3 3 5 2 3 3 3 .. -11 _ _ _ _ _ _ _ -3 -2 -1 0 1 2</strong><br />
<strong>3 5 2 3 3 3 3 .. -12 _ _ _ _ _ _ _ -4 -3 -2 -1 0 1</strong><br />
<strong>5 2 3 3 3 3 3 .. -13 _ _ _ _ _ _ _ -5 -4 -3 -2 -1 0</strong><br />
<br />
<strong>2 5 3 3 3 3 3 .. -5 -4 -3 -2 -1 0 _ _ _ _ _ _ _ 8</strong><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 />
<strong>3 3 2 5 3 3 3 .. -3 -2 -1 0 1 2 _ _ _ _ _ _ _ 10</strong><br />
<strong>3 2 5 3 3 3 3 .. -4 -3 -2 -1 0 1 _ _ _ _ _ _ _ 9</strong><br />
<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 />
<br />
<!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc4"><a name="Orwell[9], meet Porcupine[7]"></a><!-- ws:end:WikiTextHeadingRule:8 -->Orwell[9], meet Porcupine[7]</h1>
 <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 />
<ol><li>A permutation of the four large and five small steps, eg. 3 3 2 2 3 2 3 2 2<ol><li>How many of these are there? Does anyone know a formula for finding the number of possible permutations when some of the items are interchangable? Here the question is, how many permutations can we make of 4 items of Type A and 5 items of Type B?</li></ol></li><li>A scale with three step sizes: large, small, and smaller, eg. 3 2 3 2 3 2 3 3 1<ol><li>In 22edo, our &quot;smaller&quot; step is the same as our &quot;chroma&quot; (which is the interval that we alter a tone by to produce a MODMOS, L-s). However, this is not the case in larger edos! Look at <a class="wiki_link" href="/31edo">31edo</a>, where our initial scale is 4 3 4 3 4 3 4 3 3. Now our chroma is 4\31-3\31=1\31 and our &quot;smaller&quot; step is 2\31: 4 3 4 3 4 3 4 4 2! We get our &quot;smaller&quot; step by starting with s (3\31) and taking away a chroma (1\31), so we have 3\31-1\31=2\31.</li><li>So what should we call the &quot;smaller&quot; interval in our scale? Maybe some kind of diminished something-or-another?</li></ol></li><li>A scale with four steps sizes: large, small, larger and smaller, eg. 4 1 3 2 3 2 3 2 2<ol><li>This is generated by starting with L and adding a chroma, so in 22edo it's 3\22+1\22=4\22. In 31edo, that would be 4\31+1\31=5\31, and the scale in question would be 5 2 4 3 4 3 4 3 3.</li><li>So what should we call the &quot;larger&quot; step? Some kind of augmented something-or-another?</li><li>Note that in 22edo, our &quot;larger&quot; step, 4\22, is the same as two of our small steps (2\22+2\22=4\22), even though we generated our &quot;larger&quot; step by adding a chroma to a large step (3\22+1\22). In 31edo, our &quot;larger&quot; step is NOT the same as two of our small steps (4\31+1\31=5\31 does not equal 3\31+3\31=6\31)!</li></ol></li></ol><br />
So we can take advantage of the fact that two small steps in 22edo's Orwell[9] (2\22) make one &quot;larger&quot; step (4\22). If 9 tones is a few too many, we can turn some 2+2's into 4's. So for instance, the first example above goes:<br />
<br />
3 3 2 2 3 2 3 2 2<br />
3 3 4 3 2 3 4.<br />
<br />
But check it out! 3 3 4 3 2 3 4 is a MODMOS of Porcupine[7]! Here's how we can get it by chromatically-altering Porcupine[7] one tone at a time:<br />
<br />
3 3 3 3 3 3 4<br />
3 3 4 2 3 3 4<br />
3 3 4 3 2 3 4<br />
<br />
And we see, not surprisingly, that this doesn't work the same way in 31edo.<br />
<br />
Start with a MODMOS of Orwell[9]: 4 4 3 3 4 3 4 3 3<br />
Combine small steps: 4 4 6 4 3 4 6<br />
<br />
4 4 4 4 4 4 7 is as close as we can get to Porcupine[7], and it sure ain't the same. Our chroma (L-s) is 3\31, really different!<br />
4 4 7 1 4 4 7<br />
4 4 7 1 4 1 7<br />
<br />
Not even close!<br />
<br />
<!-- ws:start:WikiTextHeadingRule:10:&lt;h1&gt; --><h1 id="toc5"><a name="Names for steps"></a><!-- ws:end:WikiTextHeadingRule:10 -->Names for steps</h1>
 <br />
This is getting silly! We need better names.....<br />
<br />
So, as proposed on the page for <a class="wiki_link" href="/MODMOS%20Scales">MODMOS Scales</a>, we could call L-s a &quot;chroma&quot; and abbreviate it &quot;c&quot;. That's a good start.<br />
<br />
We have another step that's s-c, or s-(L-s) = s-L+s = 2s-L. In Porcupine[7] in 22edo, that's:<br />
s-c = 3\22-1\22 = 2\22<br />
or<br />
2s-L = 2(3\22)-4\22 = 6\22-4\22 = 2\22.<br />
Some kind of diminished step?<br />
<br />
And we have another step that's L+c, or L+(L-s) = 2L-s. In Porcupine[7] in 22edo, that's:<br />
L+c = 4\22+1\22 = 5\22<br />
or<br />
2L-s = 2(4\22)-3\22 = 8\22-3\22 = 5\22.<br />
Some kind of augmented step?<br />
<br />
Mike Battaglia proposes (at least in the case of Porcupine) &quot;d&quot; for s-c and &quot;A&quot; for L+c....<br />
<br />
<span class="commentBody">So I posted to XA: &quot;Ok, thinking it over, A and d are interesting choices to describe the additional steps, since they're so general. &quot;d&quot; is the small step minus one chroma, or s-c. But since the chroma itself is L-s, we can define d directly in terms of L a</span><span class="text_exposed_show">nd s as 2s-L. Meanwhile, &quot;A&quot; is the large step plus one chroma, or L+c. Described in terms of L and s, &quot;A&quot; is 2L-s. This allows us to quickly compute c, A, and d quickly, given L and s. As one example, Orwell[9] in 53edo has L=7 and s=5. So c=L-s=7-5=2; d=2s-L=2(5)-7=3; and A=2L-s=2(7)-5=9. Sure enough, a MODMOS of Orwell[9] with all four of these steps can be easily generated. Start with 7 5 7 5 7 5 7 5 5 and shift the second tone up by one chroma (which we computed to be 2\53), producing 9 3 7 5 7 5 7 5 5, which generalizes to AdLsLsLss! (Note that in 53edo Orwell[9] 2s=10 and A=9, not equal; while in 22edo's version of Orwell[9] 2s=4 and A=4, a potential 22edo Orwell pun.)</span><span class="commentBody"> ... ‎(Oh, and another pun is possible here, since c and d are both 1 degree in 22edo but 2 and 3 degrees, respectively, in 53edo.)&quot;</span></body></html>