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Questions about moments of symmetry (MOS)

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Author Topic: Questions about moments of symmetry (MOS)  (Read 2023 times)

Gedankenwelt

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Questions about moments of symmetry (MOS)
« on: April 23, 2012, 08:39:08 AM »

Are the following assumptions correct?
(generator g, n = #tones, unit period P = 1, no additional equivalence interval etc.)
  • If g is irrational, a scale generated by g is a MOS iff n > 1 is the denominator of a (semi-)convergent of g.
  • If g is irrational, a scale generated by g is a MOS iff it has exactly 2 step sizes.
  • If g = x/m is rational, a scale generated by g is a MOS iff n > 1 is the denominator of a (semi-)convergent of g, or n = m - 1.
  • The pseudo-Myhill property applies iff g = x/m is rational and d < n < m - 1, where d is the largest denominator of the (semi-)convergents of g.
...x and m are coprime, of course.

Thanks in advance to anyone who can help me with confirming or neglecting these assumptions! :)

P.S.: More questions may follow, and everyone is cordially invited to add questions themselves. ;)
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Mat

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Re: Questions about moments of symmetry (MOS)
« Reply #1 on: May 01, 2012, 10:02:36 AM »

Might be a question for the 'tuning' or 'tuning math' group at yahoo, this is way over my head.
Good to see you really know your theory :)
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Gedankenwelt

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Re: Questions about moments of symmetry (MOS)
« Reply #2 on: May 01, 2012, 07:20:18 PM »

Don't worry, it's much simpler than I expressed it, I'm just a little busy at the moment - sorry for that!

Convergents or semiconvergents, for example, can be easily determined when using a Stern-Brocot tree, or Erv Wilson's re-invention from it, the Scale-Tree, without any higher knowledge of math required. And "g = x/m is rational" simply means that g is the x-step interval in the tuning that divides the period into m steps. So if P represents the octave, and g = 7/12, then g is the 7-step interval in 12-EDO (the 700 cent fifth). The (semi-)convergents for g = 7/12 are 1/2, 2/3, 3/5 and 4/7, so its MOS have 2, 3, 5, 7 and m - 1 = 11 tones.

I'll add more explanations later...
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