Andrew Heathwaite's MOS Investigations: Difference between revisions

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.

This diagram is an experiment in combining the two ways I tend to visualize MOS scales -- as a chain of generators (x-axis) and as particular steps in pitch-space (y-axis).

[[MODMOS_Scales|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:

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|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:

In the Xenharmonic Alliance Facebook Group, on Dec. 1, 2011, Keenan Pepper posted a short list of MOS scales, introducing them, '<span style="">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>"

| | Porcupine[7] in 15edo

| | Porcupine[7] in 37edo

| | Porcupine[8] in 22edo

| | Neutral 3rds [7] in 17edo

| | Neutral 3rds [7] in 27edo

| | Sensi[8] in 19edo

| | Sensi[8] in 46edo

| | Sensi[8] in 27edo

| | Negri[9] in 19edo

| | Orwell[9] in 84edo

| | Orwell[9] in 53edo

| | Orwell[9] in 22edo

| | Orwell[9] in 35edo

| | Pajara[10] in 22edo

| | Blackwood[10] in 15edo

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:

<ul><li>Keenan Pepper writes about how Porcupine tempers 27/20, 15/11 and 25/18 all to the 11/8 approximation, which, he claims, is a stronger consonance than any of the intervals mentioned.</li><li>Mike Battaglia writes about how 81/80 is "tempered in" to 25/24, making it melodically useful instead of an "irritating mystery interval" which "introduces pitch drift".</li><li>MB writes about Porcupine's [[MODMOS_Scales|MODMOS]] scales (which I will deal with more below), summarizing, "<span style="">In short, when you're playing in porcupine, you should never feel like you're limited to just the 7 or 8-note MOS. Just freeform modify notes by L-s as much as you want, deliberately, in a willful attempt to explore porcupine chromaticism. It's even easier than meantone chromaticism.</span>"</li><li>MB: "I<span style="">n porcupine, bIII/bIII/bIII = IV/IV. This is the same thing as saying that 6/5 * 6/5 * 6/5 = 4/3 * 4/3</span>."</li><li>Someone argues that Porcupine doesn't do that great in the 5-limit after all, saying, "<span style="">Its only real selling-point over optimal meantone is simpler 7-limit and 11-limit approximations, but that assumes that these are a good in their own right and thus worth sacrificing some 5-limit efficiency; for anyone other than a dyed-in-the-wool xenharmonist, that's a questionable assumption to make.</span>" (As for me, I want those 7- and 11-limit approximations, and I could care less about a 5-limit temperament to rival meantone. I don't compose in 5-limit temperaments, period.)</li><li>In response to the above, Keenan Pepper says, "<span style="">You mentioned that almost every interval in the diatonic scale is a 9-limit consonance? Well, every interval in porcupine[7] is an 11-limit consonance! 1/1 10/9 9/8 6/5 5/4 4/3 11/8 16/11 3/2 8/5 5/3 16/9 9/5 2/1. Bam!</span>" (This is relevant to my work, which assumes composers want 11-limit approximations.)</li><li>I (Andrew Heathwaite) added, "<span style="">...maybe another description for what Porcupine is good for is a *gateway* from 5 and 7 to 11, for those comfortable with the former and curious about the latter. As a full 11-limit temperament, it is efficient and easy.</span>"</li></ul>

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.

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: