Maximal evenness
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Within every [[edo]] one can specify a "maximally even" (ME) or "quasi-equal" scale for every smaller edo. The maximally even scale is the closest the parent edo can get to representing the smaller edo. 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. (a) and (b) above imply that the ME scale will be a [[MOSScales|moment of symmetry scale]]. For instance, here are all the ME scales available in [[31edo]]: <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 2 .. 15 16</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 3 .. 10 10 11</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 4 .. 8 8 8 7</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 5 .. 6 6 6 6 7</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 6 .. 5 5 5 5 5 6</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 7 .. 5 4 5 4 5 4 4</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 8 .. 4 4 4 4 4 4 4 3</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 9 .. 4 3 4 3 4 3 4 3 3</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 10 . 3 3 3 3 3 3 3 3 3 4</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 11 . 2 3 3 3 3 2 3 3 3 3 3</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 12 . 3 3 2 3 2 3 3 2 3 2 3 2</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 13 . 3 2 3 2 2 3 2 3 2 2 3 2 2</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 14 . 2 2 2 2 3 2 2 2 2 3 2 2 2 3</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 15 . 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 16 . 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 17 . 2 2 2 2 2 1 2 2 2 2 2 1 2 2 2 2 1</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 18 . 2 1 2 2 1 2 2 2 1 2 2 1 2 2 2 1 2 2</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 19 . 2 1 2 1 2 2 1 2 2 1 2 1 2 2 1 2 2 1 2</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 20 . 2 2 1 2 1 2 1 2 1 2 2 1 2 1 2 1 2 1 2 1</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 21 . 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 2</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 22 . 1 2 1 1 2 1 2 1 1 2 1 2 1 1 2 1 2 1 1 2 1 2</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 23 . 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 24 . 1 2 1 1 1 2 1 1 2 1 1 1 2 1 1 2 1 1 1 2 1 1 2 1</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 25 . 1 2 1 1 1 2 1 1 1 2 1 1 1 2 1 1 1 2 1 1 1 2 1 1 1</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 26 . 2 1 1 1 1 2 1 1 1 1 2 1 1 1 1 2 1 1 1 1 2 1 1 1 1 1</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 27 . 2 1 1 1 1 1 1 2 1 1 1 1 1 1 2 1 1 1 1 1 1 2 1 1 1 1 1</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 28 . 2 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 29 . 2 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 30 . 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 2</span> And here are all the ME scales available in [[13edo]]: <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;">2 .. 6 7</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 3 .. 4 4 5</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 4 .. 3 3 3 4</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 5 .. 2 3 2 3 3</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 6 .. 2 2 2 2 2 3</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 7 .. 2 2 2 2 2 2 1</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 8 .. 2 2 1 2 2 1 2 1</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 9 .. 2 1 2 1 2 1 2 1 1</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 10 . 2 1 1 2 1 1 2 1 1 1</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 11 . 2 1 1 1 1 2 1 1 1 1 1</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 12 . 1 1 1 1 1 1 1 1 1 1 1 2 </span> The ME scales in 31edo will be closer to equal than those in 13edo, since the two step sizes used to approximate equal will differ by a smaller interval (one 31th of an octave instead of one 13th). The parent edo will better represent smaller edos than larger ones. With edos larger than 1/2 of the parent edo, the step sizes will be 2 and 1, which are, proportionally speaking, far from equal. So 13edo's 3 3 3 4 will sound more like 4edo than its 1 1 1 1 1 1 1 1 1 1 1 2 will sound like 12edo.
Original HTML content:
<html><head><title>Maximal evenness</title></head><body>Within every <a class="wiki_link" href="/edo">edo</a> one can specify a "maximally even" (ME) or "quasi-equal" scale for every smaller edo. The maximally even scale is the closest the parent edo can get to representing the smaller edo.<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 /> (a) and (b) above imply that the ME scale will be a <a class="wiki_link" href="/MOSScales">moment of symmetry scale</a>.<br /> <br /> For instance, here are all the ME scales available in <a class="wiki_link" href="/31edo">31edo</a>:<br /> <br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 2 .. 15 16</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 3 .. 10 10 11</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 4 .. 8 8 8 7</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 5 .. 6 6 6 6 7</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 6 .. 5 5 5 5 5 6</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 7 .. 5 4 5 4 5 4 4</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 8 .. 4 4 4 4 4 4 4 3</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 9 .. 4 3 4 3 4 3 4 3 3</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 10 . 3 3 3 3 3 3 3 3 3 4</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 11 . 2 3 3 3 3 2 3 3 3 3 3</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 12 . 3 3 2 3 2 3 3 2 3 2 3 2</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 13 . 3 2 3 2 2 3 2 3 2 2 3 2 2</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 14 . 2 2 2 2 3 2 2 2 2 3 2 2 2 3</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 15 . 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 16 . 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 17 . 2 2 2 2 2 1 2 2 2 2 2 1 2 2 2 2 1</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 18 . 2 1 2 2 1 2 2 2 1 2 2 1 2 2 2 1 2 2</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 19 . 2 1 2 1 2 2 1 2 2 1 2 1 2 2 1 2 2 1 2</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 20 . 2 2 1 2 1 2 1 2 1 2 2 1 2 1 2 1 2 1 2 1</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 21 . 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 2</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 22 . 1 2 1 1 2 1 2 1 1 2 1 2 1 1 2 1 2 1 1 2 1 2</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 23 . 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 24 . 1 2 1 1 1 2 1 1 2 1 1 1 2 1 1 2 1 1 1 2 1 1 2 1</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 25 . 1 2 1 1 1 2 1 1 1 2 1 1 1 2 1 1 1 2 1 1 1 2 1 1 1</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 26 . 2 1 1 1 1 2 1 1 1 1 2 1 1 1 1 2 1 1 1 1 2 1 1 1 1 1</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 27 . 2 1 1 1 1 1 1 2 1 1 1 1 1 1 2 1 1 1 1 1 1 2 1 1 1 1 1</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 28 . 2 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 29 . 2 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 30 . 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 2</span><br /> <br /> And here are all the ME scales available in <a class="wiki_link" href="/13edo">13edo</a>:<br /> <br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;">2 .. 6 7</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 3 .. 4 4 5</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 4 .. 3 3 3 4</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 5 .. 2 3 2 3 3</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 6 .. 2 2 2 2 2 3</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 7 .. 2 2 2 2 2 2 1</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 8 .. 2 2 1 2 2 1 2 1</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 9 .. 2 1 2 1 2 1 2 1 1</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 10 . 2 1 1 2 1 1 2 1 1 1</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 11 . 2 1 1 1 1 2 1 1 1 1 1</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 12 . 1 1 1 1 1 1 1 1 1 1 1 2 </span><br /> <br /> The ME scales in 31edo will be closer to equal than those in 13edo, since the two step sizes used to approximate equal will differ by a smaller interval (one 31th of an octave instead of one 13th).<br /> <br /> The parent edo will better represent smaller edos than larger ones. With edos larger than 1/2 of the parent edo, the step sizes will be 2 and 1, which are, proportionally speaking, far from equal. So 13edo's 3 3 3 4 will sound more like 4edo than its 1 1 1 1 1 1 1 1 1 1 1 2 will sound like 12edo.</body></html>