Mathematics of MOS: Difference between revisions

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== Properties ==

~~=== Basic properties ===~~

* Every MOS scale has two ''child MOS'' scales. The two children of the MOS scale aL bs are (a + b)L as (corresponding to soft-of-basic aLbs) and aL (a + b)s (corresponding to hard-of-basic aLbs).

* Every MOS scale (with a specified [[equave]] ''E''), excluding aL as⟨''E''⟩, has a ''parent MOS''. If a > b, the parent of aLbs is min(a, b)L|a − b|s; if a < b, the parent of aLbs is |a − b|L min(a, b)s.

~~=== Advanced discussion ===~~

Let us represent the period as 1. This would be the logarithm base 2 of 2 if the period is an octave, or in general we can measure intervals by the log base P when P is the period. Suppose the fractions a/b and c/d are a [[Wikipedia:Farey_sequence#Farey_neighbours|Farey pair]], meaning that a/b < c/d and bc - ad = 1. If g = (1-t)(a/b) + t(c/d) for 0 ? t ? 1, then when t = 0, the scale generated by g will consist of an equal division of 1 (representing P) into steps of size 1/b, and when t = 1 into steps of size 1/d. In between, when t = b/(b + d), we obtain a generator equal to the [[Wikipedia:Mediant_%28mathematics%29|mediant]] (a + c)/(b + d) and which will divide the period into b+d equal steps. For all other values a/b < g < c/d we obtain two different sizes of steps, the small steps s, and the large steps L, with the total number of steps b+d, and these scales are the MOS associated to the Farey pair. When g is between a/b and (a + c)/(b + d) there will be b large steps and d small steps, and when it is between (a + c)/(b + d) and c/d, d large steps and b small ones.

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 == Properties ==
-=== Basic properties ===
-* Every MOS scale has two ''child MOS'' scales. The two children of the MOS scale aL bs are (a + b)L as (corresponding to soft-of-basic aLbs) and aL (a + b)s (corresponding to hard-of-basic aLbs).
-* Every MOS scale (with a specified [[equave]] ''E''), excluding aL as⟨''E''⟩, has a ''parent MOS''. If a > b, the parent of aLbs is min(a, b)L|a &minus; b|s; if a < b, the parent of aLbs is |a &minus; b|L min(a, b)s.
-=== Advanced discussion ===
 Let us represent the period as 1. This would be the logarithm base 2 of 2 if the period is an octave, or in general we can measure intervals by the log base P when P is the period. Suppose the fractions a/b and c/d are a [[Wikipedia:Farey_sequence#Farey_neighbours|Farey pair]], meaning that a/b < c/d and bc - ad = 1. If g = (1-t)(a/b) + t(c/d) for 0 ? t ? 1, then when t = 0, the scale generated by g will consist of an equal division of 1 (representing P) into steps of size 1/b, and when t = 1 into steps of size 1/d. In between, when t = b/(b + d), we obtain a generator equal to the [[Wikipedia:Mediant_%28mathematics%29|mediant]] (a + c)/(b + d) and which will divide the period into b+d equal steps. For all other values a/b < g < c/d we obtain two different sizes of steps, the small steps s, and the large steps L, with the total number of steps b+d, and these scales are the MOS associated to the Farey pair. When g is between a/b and (a + c)/(b + d) there will be b large steps and d small steps, and when it is between (a + c)/(b + d) and c/d, d large steps and b small ones.