User:Arseniiv/Infinite MOS: Difference between revisions

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''(Also this page contains mentions of me which isn’t very encyclopedic but I wonder how do I restate them in a neutral way without working on clarifying those places instead, which I have no time for, right now.)''

An '''infinite''' (or '''aperiodic''' or '''quasiperiodic''') '''MOS scale'''{{idiosyncratic}} is a generalization of a [[MOS scale]] to aperiodic scales. Mathematically, an infinite MOS pattern is a [[wikipedia:Sturmian word]] (but extended in both directions), like a periodic MOS pattern is a Christoffel word.

An '''infinite''' (or '''aperiodic''' or '''quasiperiodic''') '''MOS scale'''{{idiosyncratic}} is a generalization of a [[MOS scale]] to aperiodic scales. Mathematically, an infinite MOS pattern is a [[wikipedia:Sturmian word|Sturmian word]] (but extended in both directions), like a periodic MOS pattern is a Christoffel word.

== Properties ==

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== Definition ==

To get a scale pattern, one just uses a Sturmian word by using any of its [~~https~~:~~//en.wikipedia.org/wiki/~~Sturmian_word#Definition definitions] which allows extension in both directions, for example as a cutting sequence of a straight line. Then step sizes are specified. As the scale is aperiodic, step sizes (and hardness) are divorced from the structure of the pattern, and “step abundance” (the slope of a cutting line) is its own variable.

To get a scale pattern, one just uses a Sturmian word by using any of its [[wikipedia:Sturmian_word#Definition|definitions]] which allows extension in both directions, for example as a cutting sequence of a straight line. Then step sizes are specified. As the scale is aperiodic, step sizes (and hardness) are divorced from the structure of the pattern, and “step abundance” (the slope of a cutting line) is its own variable.

When step abundance gets exactly rational, a Sturmian word becomes an infinitely repeated Christoffel word (and for some reason there doesn’t seem to exist a general term for this kind of cutting word; “Sturmian” is deemed to apply only to irrational slopes) and the scale becomes the usual periodic MOS scale. So, an aperiodic MOS scale is close to infinitely many periodic MOS scales, with periods larger and larger.

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 ''(Also this page contains mentions of me which isn’t very encyclopedic but I wonder how do I restate them in a neutral way without working on clarifying those places instead, which I have no time for, right now.)''
-An '''infinite''' (or '''aperiodic''' or '''quasiperiodic''') '''MOS scale'''{{idiosyncratic}} is a generalization of a [[MOS scale]] to aperiodic scales. Mathematically, an infinite MOS pattern is a [[wikipedia:Sturmian word]] (but extended in both directions), like a periodic MOS pattern is a Christoffel word.
+An '''infinite''' (or '''aperiodic''' or '''quasiperiodic''') '''MOS scale'''{{idiosyncratic}} is a generalization of a [[MOS scale]] to aperiodic scales. Mathematically, an infinite MOS pattern is a [[wikipedia:Sturmian word|Sturmian word]] (but extended in both directions), like a periodic MOS pattern is a Christoffel word.
 == Properties ==
@@ Line 19: / Line 19: @@
 == Definition ==
-To get a scale pattern, one just uses a Sturmian word by using any of its [https://en.wikipedia.org/wiki/Sturmian_word#Definition definitions] which allows extension in both directions, for example as a cutting sequence of a straight line. Then step sizes are specified. As the scale is aperiodic, step sizes (and hardness) are divorced from the structure of the pattern, and “step abundance” (the slope of a cutting line) is its own variable.
+To get a scale pattern, one just uses a Sturmian word by using any of its [[wikipedia:Sturmian_word#Definition|definitions]] which allows extension in both directions, for example as a cutting sequence of a straight line. Then step sizes are specified. As the scale is aperiodic, step sizes (and hardness) are divorced from the structure of the pattern, and “step abundance” (the slope of a cutting line) is its own variable.
 When step abundance gets exactly rational, a Sturmian word becomes an infinitely repeated Christoffel word (and for some reason there doesn’t seem to exist a general term for this kind of cutting word; “Sturmian” is deemed to apply only to irrational slopes) and the scale becomes the usual periodic MOS scale. So, an aperiodic MOS scale is close to infinitely many periodic MOS scales, with periods larger and larger.