Andrew Heathwaite's MOS Investigations

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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.

==Expanding on "Maximal Evenness"== 
"[[Maximal Evenness]]" (ME, aka "Quasi-Equalness," QE) is a quality certain MOS scales within equal scales can have.

The maximally even scale will be one:
a. which contains exactly two step sizes as close in size as possible (differing by exactly one degree of the parent edo).
b. whose steps are distributed as evenly as possible.

For every n-edo, there are ME scales for every number of tones t where 1 < t < n.

In an ME scale, L and s differ by exactly one degree of the parent edo. So now I'm wondering about scales which differ by //two// degrees of the parent scale. I'm going to examine [[MOS Scales of 37edo]] to see if I can find scales of this type. I'll call it ME(2) for now, to mean something like, "Maximally Even, given that the difference between L and s must be 2 degrees". I was surprised to find ME(2) scales for every //odd number// of tones t where 1< t < 37 -- and none for any even numbers.

ME(2) for 3 tones: 13\37: 13 13 11
ME(2) for 5 tones: 7\37: 7 7 7 7 9
ME(2) for 7 tones: 5\37: 5 5 5 5 5 5 7
ME(2) for 9 tones: 8\37: 3 5 3 5 3 5 3 5 5
ME(2) for 11 tones: 17\37: 5 3 3 3 3 5 3 3 3 3 3
ME(2) for 13 tones: 3\37: 3 3 3 3 3 3 3 3 3 3 3 3 1
ME(2) for 15 tones: 10\37: 3 3 3 1 3 3 3 1 3 3 3 1 3 3 1
ME(2) for 17 tones: 11\37: 3 3 1 3 1 3 3 1 3 1 3 3 1 3 1 3 1
ME(2) for 19 tones: 4\37: 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 1
ME(2) for 21 tones: 14\37: 1 1 3 1 1 3 1 3 1 1 3 1 1 3 1 3 1 1 3 1 3
ME(2) for 23 tones: 16\37: 1 1 1 3 1 1 3 1 1 3 1 1 1 3 1 1 3 1 1 3 1 1 3
ME(2) for 25 tones: 6\37: 3 1 1 1 3 1 1 1 3 1 1 1 3 1 1 1 3 1 1 1 3 1 1 1 1
ME(2) for 27 tones: 15\37: 1 1 1 1 1 3 1 1 1 1 3 1 1 1 1 1 3 1 1 1 1 3 1 1 1 1 3
ME(2) for 29 tones: 9\37: 3 1 1 1 1 1 1 3 1 1 1 1 1 1 3 1 1 1 1 1 1 3 1 1 1 1 1 1 1
ME(2) for 31 tones: 12\37: 3 1 1 1 1 1 1 1 1 1 3 1 1 1 1 1 1 1 1 1 3 1 1 1 1 1 1 1 1 1 1
ME(2) for 33 tones: 18\37 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
ME(2) for 35 tones: 1\37: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3

Ordinary ME scales eventually break down into L:s = 2:1 (not very "even" at all). Likewise, ME(2) scales eventually break down into L:s = 3:1, also not very even. But before 3:1 there's 5:3, before that 7:5, before that 9:7, etc. Interesting that there are no ME(2) scales with an even number of tones -- and interesting that in the ME(2) scales, all steps are odd numbers of degrees in size!

So now I'll look for ME(3) scales in 37edo. Although I'm generalizing the "Maximal Evenness" idea, it's quite clear that these scales are not necessarily "even" and that a better name is needed. I ain't got one yet.

It turns out that there are no ME(3) scales for numbers of tones that are divisible by 3! I also didn't see any for t = 20, 23, 26, 29, 32 or 35 -- and I have no idea why that would be!

ME(3) for 2 tones: 17\37: 17 20
ME(3) for 4 tones: 10\37: 10 10 10 7
ME(3) for 5 tones: 8\37: 8 8 8 8 5
ME(3) for 7 tones: 11\37: 7 4 7 4 7 4 4
ME(3) for 8 tones: 5\37: 5 5 5 5 5 5 5 2
ME(3) for 10 tones: 4\37: 4 4 4 4 4 4 4 4 4 1
ME(3) for 11 tones: 7\37: 5 2 5 2 5 2 5 2 5 2 2
ME(3) for 13 tones: 14\37: 1 4 1 4 4 1 4 1 4 4 1 4 4
ME(3) for 14 tones: 13\37: 2 2 2 2 5 2 2 2 2 5 2 2 2 5
ME(3) for 16 tones: 16\37: 1 1 4 1 4 1 4 1 1 4 1 4 1 4 1 4
ME(3) for 17 tones: 2\37: 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 5
ME(3) for 19 tones: 6\37: 4 1 1 4 1 1 4 1 1 4 1 1 4 1 1 4 1 1 1
ME(3) for 20 tones: I don't see one!
ME(3) for 22 tones: 15\37: 1 1 1 1 4 1 1 1 4 1 1 1 1 4 1 1 1 4 1 1 1 4
ME(3) for 23 tones: I don't see one!
ME(3) for 25 tones: 9\37: 4 1 1 1 1 1 4 1 1 1 1 1 4 1 1 1 1 1 4 1 1 1 1 1 1
ME(3) for 26 tones: I don't see one!
ME(3) for 28 tones: 12\37: 4 1 1 1 1 1 1 1 1 4 1 1 1 1 1 1 1 1 4 1 1 1 1 1 1 1 1 1
ME(3) for 29 tones: I don't see one!
ME(3) for 31 tones: 18\37: 4 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
ME(3) for 32 tones: I don't see one!
ME(3) for 34 tones: 1\37: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4
ME(3) for 35 tones: I don't see one!

So that's where I'm leaving this problem for now.

==MOS Scales with similar generators== 
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.

[[image:137edo_MOS_031_demo_correction.png]]
[[image:137edo_MOS_032_demo.png]]

Update: I decided to go with [[127edo]] and have completed the visual study. See [[MOS Scales of 127edo]].

==Notes on Keenan Pepper's Diatonic-like MOS Scales== 

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>"

This sounds interesting. I'm using this space to take some notes on the scales he lists:

||~ Scale Name ||~ Generator ||~ L ||~ s ||~ c ||~ L:s ||~ s:c ||
|| Porcupine[7] in 15edo || 160 || 240 || 160 || 80 || 2:1 = 2 || 2:1 = 2 ||
|| Porcupine[7] in 37edo || 162.16 || 227.03 || 162.16 || 64.87 || 7:5 = 1.4 || 5:2 = 2.5 ||
|| Porcupine[8] in 22edo || 163.64 || 212.18 || 163.64 || 54.55 || 4:3 = 1.33 || 3:1 = 3 ||
|| Neutral 3rds [7] in 17edo || 352.94 || 211.77 || 141.18 || 70.59 || 3:2 = 1.5 || 2:1 = 2 ||
|| Neutral 3rds [7] in 27edo || 355.56 || 222.22 || 133.33 || 88.89 || 5:3 = 1.67 || 3:2 = 1.5 ||
|| Sensi[8] in 19edo || 442.11 || 189.47 || 126.32 || 63.16 || 3:2 = 1.5 || 2:1 = 2 ||
|| Sensi[8] in 46edo || 443.48 || 182.61 || 130.44 || 52.17 || 7:5 = 1.4 || 5:2 = 2.5 ||
|| Sensi[8] in 27edo || 444.44 || 177.78 || 133.33 || 44.44 || 4:3 = 1.33 || 3:1 = 3 ||
|| Negri[9] in 19edo || 126.32 || 189.47 || 126.32 || 63.16 || 3:2 = 1.5 || 2:1 = 2 ||
|| Orwell[9] in 84edo || 271.43 || 157.14 || 114.29 || 42.86 || 11:8 = 1.38 || 8:3 = 2.67 ||
|| Orwell[9] in 53edo || 271.70 || 158.49 || 113.2 || 45.28 || 7:5 = 1.4 || 5:2 = 2.5 ||
|| Orwell[9] in 22edo || 272.73 || 163.64 || 109.09 || 54.55 || 3:2 = 1.5 || 2:1 = 2 ||
|| Orwell[9] in 35edo || 274.29 || 171.43 || 102.86 || 68.57 || 5:3 = 1.67 || 3:2 = 1.5 ||
|| Pajara[10] in 22edo || 109.09 || 163.64 || 109.09 || 54.55 || 3:2 = 1.5 || 2:1 = 2 ||
|| Blackwood[10] in 15edo || 80 || 160 || 80 || - || 2:1 = 2 ||   ||


=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:

[[image:porcupine_mos_overview_140edo.jpg]]

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 //there is no standard form for Porcupine[22]//. This means that, after a certain point, we have to //pick a tuning// (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.

==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] 6|0 #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] 6|0 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] 6|0 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>

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<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 />
<!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-Expanding on &quot;Maximal Evenness&quot;"></a><!-- ws:end:WikiTextHeadingRule:0 -->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 />
<br />
The maximally even scale will be one:<br />
a. which contains exactly two step sizes as close in size as possible (differing by exactly one degree of the parent edo).<br />
b. whose steps are distributed as evenly as possible.<br />
<br />
For every n-edo, there are ME scales for every number of tones t where 1 &lt; t &lt; n.<br />
<br />
In an ME scale, L and s differ by exactly one degree of the parent edo. So now I'm wondering about scales which differ by <em>two</em> degrees of the parent scale. I'm going to examine <a class="wiki_link" href="/MOS%20Scales%20of%2037edo">MOS Scales of 37edo</a> to see if I can find scales of this type. I'll call it ME(2) for now, to mean something like, &quot;Maximally Even, given that the difference between L and s must be 2 degrees&quot;. I was surprised to find ME(2) scales for every <em>odd number</em> of tones t where 1&lt; t &lt; 37 -- and none for any even numbers.<br />
<br />
ME(2) for 3 tones: 13\37: 13 13 11<br />
ME(2) for 5 tones: 7\37: 7 7 7 7 9<br />
ME(2) for 7 tones: 5\37: 5 5 5 5 5 5 7<br />
ME(2) for 9 tones: 8\37: 3 5 3 5 3 5 3 5 5<br />
ME(2) for 11 tones: 17\37: 5 3 3 3 3 5 3 3 3 3 3<br />
ME(2) for 13 tones: 3\37: 3 3 3 3 3 3 3 3 3 3 3 3 1<br />
ME(2) for 15 tones: 10\37: 3 3 3 1 3 3 3 1 3 3 3 1 3 3 1<br />
ME(2) for 17 tones: 11\37: 3 3 1 3 1 3 3 1 3 1 3 3 1 3 1 3 1<br />
ME(2) for 19 tones: 4\37: 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 1<br />
ME(2) for 21 tones: 14\37: 1 1 3 1 1 3 1 3 1 1 3 1 1 3 1 3 1 1 3 1 3<br />
ME(2) for 23 tones: 16\37: 1 1 1 3 1 1 3 1 1 3 1 1 1 3 1 1 3 1 1 3 1 1 3<br />
ME(2) for 25 tones: 6\37: 3 1 1 1 3 1 1 1 3 1 1 1 3 1 1 1 3 1 1 1 3 1 1 1 1<br />
ME(2) for 27 tones: 15\37: 1 1 1 1 1 3 1 1 1 1 3 1 1 1 1 1 3 1 1 1 1 3 1 1 1 1 3<br />
ME(2) for 29 tones: 9\37: 3 1 1 1 1 1 1 3 1 1 1 1 1 1 3 1 1 1 1 1 1 3 1 1 1 1 1 1 1<br />
ME(2) for 31 tones: 12\37: 3 1 1 1 1 1 1 1 1 1 3 1 1 1 1 1 1 1 1 1 3 1 1 1 1 1 1 1 1 1 1<br />
ME(2) for 33 tones: 18\37 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1<br />
ME(2) for 35 tones: 1\37: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3<br />
<br />
Ordinary ME scales eventually break down into L:s = 2:1 (not very &quot;even&quot; at all). Likewise, ME(2) scales eventually break down into L:s = 3:1, also not very even. But before 3:1 there's 5:3, before that 7:5, before that 9:7, etc. Interesting that there are no ME(2) scales with an even number of tones -- and interesting that in the ME(2) scales, all steps are odd numbers of degrees in size!<br />
<br />
So now I'll look for ME(3) scales in 37edo. Although I'm generalizing the &quot;Maximal Evenness&quot; idea, it's quite clear that these scales are not necessarily &quot;even&quot; and that a better name is needed. I ain't got one yet.<br />
<br />
It turns out that there are no ME(3) scales for numbers of tones that are divisible by 3! I also didn't see any for t = 20, 23, 26, 29, 32 or 35 -- and I have no idea why that would be!<br />
<br />
ME(3) for 2 tones: 17\37: 17 20<br />
ME(3) for 4 tones: 10\37: 10 10 10 7<br />
ME(3) for 5 tones: 8\37: 8 8 8 8 5<br />
ME(3) for 7 tones: 11\37: 7 4 7 4 7 4 4<br />
ME(3) for 8 tones: 5\37: 5 5 5 5 5 5 5 2<br />
ME(3) for 10 tones: 4\37: 4 4 4 4 4 4 4 4 4 1<br />
ME(3) for 11 tones: 7\37: 5 2 5 2 5 2 5 2 5 2 2<br />
ME(3) for 13 tones: 14\37: 1 4 1 4 4 1 4 1 4 4 1 4 4<br />
ME(3) for 14 tones: 13\37: 2 2 2 2 5 2 2 2 2 5 2 2 2 5<br />
ME(3) for 16 tones: 16\37: 1 1 4 1 4 1 4 1 1 4 1 4 1 4 1 4<br />
ME(3) for 17 tones: 2\37: 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 5<br />
ME(3) for 19 tones: 6\37: 4 1 1 4 1 1 4 1 1 4 1 1 4 1 1 4 1 1 1<br />
ME(3) for 20 tones: I don't see one!<br />
ME(3) for 22 tones: 15\37: 1 1 1 1 4 1 1 1 4 1 1 1 1 4 1 1 1 4 1 1 1 4<br />
ME(3) for 23 tones: I don't see one!<br />
ME(3) for 25 tones: 9\37: 4 1 1 1 1 1 4 1 1 1 1 1 4 1 1 1 1 1 4 1 1 1 1 1 1<br />
ME(3) for 26 tones: I don't see one!<br />
ME(3) for 28 tones: 12\37: 4 1 1 1 1 1 1 1 1 4 1 1 1 1 1 1 1 1 4 1 1 1 1 1 1 1 1 1<br />
ME(3) for 29 tones: I don't see one!<br />
ME(3) for 31 tones: 18\37: 4 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1<br />
ME(3) for 32 tones: I don't see one!<br />
ME(3) for 34 tones: 1\37: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4<br />
ME(3) for 35 tones: I don't see one!<br />
<br />
So that's where I'm leaving this problem for now.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="x-MOS Scales with similar generators"></a><!-- ws:end:WikiTextHeadingRule:2 -->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 />
<br />
<!-- ws:start:WikiTextLocalImageRule:318:&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:318 --><br />
<!-- ws:start:WikiTextLocalImageRule:319:&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:319 --><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 />
<br />
<!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="x-Notes on Keenan Pepper's Diatonic-like MOS Scales"></a><!-- ws:end:WikiTextHeadingRule:4 -->Notes on Keenan Pepper's Diatonic-like MOS Scales</h2>
 <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 />
<br />
This sounds interesting. I'm using this space to take some notes on the scales he lists:<br />
<br />


<table class="wiki_table">
    <tr>
        <th>Scale Name<br />
</th>
        <th>Generator<br />
</th>
        <th>L<br />
</th>
        <th>s<br />
</th>
        <th>c<br />
</th>
        <th>L:s<br />
</th>
        <th>s:c<br />
</th>
    </tr>
    <tr>
        <td>Porcupine[7] in 15edo<br />
</td>
        <td>160<br />
</td>
        <td>240<br />
</td>
        <td>160<br />
</td>
        <td>80<br />
</td>
        <td>2:1 = 2<br />
</td>
        <td>2:1 = 2<br />
</td>
    </tr>
    <tr>
        <td>Porcupine[7] in 37edo<br />
</td>
        <td>162.16<br />
</td>
        <td>227.03<br />
</td>
        <td>162.16<br />
</td>
        <td>64.87<br />
</td>
        <td>7:5 = 1.4<br />
</td>
        <td>5:2 = 2.5<br />
</td>
    </tr>
    <tr>
        <td>Porcupine[8] in 22edo<br />
</td>
        <td>163.64<br />
</td>
        <td>212.18<br />
</td>
        <td>163.64<br />
</td>
        <td>54.55<br />
</td>
        <td>4:3 = 1.33<br />
</td>
        <td>3:1 = 3<br />
</td>
    </tr>
    <tr>
        <td>Neutral 3rds [7] in 17edo<br />
</td>
        <td>352.94<br />
</td>
        <td>211.77<br />
</td>
        <td>141.18<br />
</td>
        <td>70.59<br />
</td>
        <td>3:2 = 1.5<br />
</td>
        <td>2:1 = 2<br />
</td>
    </tr>
    <tr>
        <td>Neutral 3rds [7] in 27edo<br />
</td>
        <td>355.56<br />
</td>
        <td>222.22<br />
</td>
        <td>133.33<br />
</td>
        <td>88.89<br />
</td>
        <td>5:3 = 1.67<br />
</td>
        <td>3:2 = 1.5<br />
</td>
    </tr>
    <tr>
        <td>Sensi[8] in 19edo<br />
</td>
        <td>442.11<br />
</td>
        <td>189.47<br />
</td>
        <td>126.32<br />
</td>
        <td>63.16<br />
</td>
        <td>3:2 = 1.5<br />
</td>
        <td>2:1 = 2<br />
</td>
    </tr>
    <tr>
        <td>Sensi[8] in 46edo<br />
</td>
        <td>443.48<br />
</td>
        <td>182.61<br />
</td>
        <td>130.44<br />
</td>
        <td>52.17<br />
</td>
        <td>7:5 = 1.4<br />
</td>
        <td>5:2 = 2.5<br />
</td>
    </tr>
    <tr>
        <td>Sensi[8] in 27edo<br />
</td>
        <td>444.44<br />
</td>
        <td>177.78<br />
</td>
        <td>133.33<br />
</td>
        <td>44.44<br />
</td>
        <td>4:3 = 1.33<br />
</td>
        <td>3:1 = 3<br />
</td>
    </tr>
    <tr>
        <td>Negri[9] in 19edo<br />
</td>
        <td>126.32<br />
</td>
        <td>189.47<br />
</td>
        <td>126.32<br />
</td>
        <td>63.16<br />
</td>
        <td>3:2 = 1.5<br />
</td>
        <td>2:1 = 2<br />
</td>
    </tr>
    <tr>
        <td>Orwell[9] in 84edo<br />
</td>
        <td>271.43<br />
</td>
        <td>157.14<br />
</td>
        <td>114.29<br />
</td>
        <td>42.86<br />
</td>
        <td>11:8 = 1.38<br />
</td>
        <td>8:3 = 2.67<br />
</td>
    </tr>
    <tr>
        <td>Orwell[9] in 53edo<br />
</td>
        <td>271.70<br />
</td>
        <td>158.49<br />
</td>
        <td>113.2<br />
</td>
        <td>45.28<br />
</td>
        <td>7:5 = 1.4<br />
</td>
        <td>5:2 = 2.5<br />
</td>
    </tr>
    <tr>
        <td>Orwell[9] in 22edo<br />
</td>
        <td>272.73<br />
</td>
        <td>163.64<br />
</td>
        <td>109.09<br />
</td>
        <td>54.55<br />
</td>
        <td>3:2 = 1.5<br />
</td>
        <td>2:1 = 2<br />
</td>
    </tr>
    <tr>
        <td>Orwell[9] in 35edo<br />
</td>
        <td>274.29<br />
</td>
        <td>171.43<br />
</td>
        <td>102.86<br />
</td>
        <td>68.57<br />
</td>
        <td>5:3 = 1.67<br />
</td>
        <td>3:2 = 1.5<br />
</td>
    </tr>
    <tr>
        <td>Pajara[10] in 22edo<br />
</td>
        <td>109.09<br />
</td>
        <td>163.64<br />
</td>
        <td>109.09<br />
</td>
        <td>54.55<br />
</td>
        <td>3:2 = 1.5<br />
</td>
        <td>2:1 = 2<br />
</td>
    </tr>
    <tr>
        <td>Blackwood[10] in 15edo<br />
</td>
        <td>80<br />
</td>
        <td>160<br />
</td>
        <td>80<br />
</td>
        <td>-<br />
</td>
        <td>2:1 = 2<br />
</td>
        <td><br />
</td>
    </tr>
</table>

<br />
<br />
<!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name="Porcupine Temperament"></a><!-- ws:end:WikiTextHeadingRule:6 -->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:8:&lt;h1&gt; --><h1 id="toc4"><a name="Porcupine Chromaticism"></a><!-- ws:end:WikiTextHeadingRule:8 -->Porcupine Chromaticism</h1>
 <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 />
<!-- ws:start:WikiTextLocalImageRule:320:&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:320 --><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 />
<br />
<!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc5"><a name="Porcupine Chromaticism-Modes of Porcupine[7]"></a><!-- ws:end:WikiTextHeadingRule:10 -->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:12:&lt;h2&gt; --><h2 id="toc6"><a name="Porcupine Chromaticism-Modes of Porcupine[7] that have one chromatic alteration"></a><!-- ws:end:WikiTextHeadingRule:12 -->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] 6|0 #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] 6|0 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] 6|0 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:14:&lt;h1&gt; --><h1 id="toc7"><a name="Orwell[9], meet Porcupine[7]"></a><!-- ws:end:WikiTextHeadingRule:14 -->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:16:&lt;h1&gt; --><h1 id="toc8"><a name="Names for steps"></a><!-- ws:end:WikiTextHeadingRule:16 -->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>