Mathematics of MOS: Difference between revisions
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Yet another way of classifying MOS is via {{w|Minkowski's question-mark function}}. Here ?(''x''), {{nowrap|? : ℝ → R}} is a continuously increasing function which has some peculiar properties, one being that it sends rational numbers to {{w|dyadic rational}}s. Hence if ''q'' is a rational number {{nowrap|0 < ''q'' < 1}} in use in the mediant system of classifying MOS scales, {{nowrap|''r'' {{=}} ?(''q'')}} = {{sfrac|''A''|2<sup>''n''</sup>}} will be a dyadic rational number which can also be used. Note that the ? function is invertible, and it and its inverse function, the Box function, have code given for them in the algorithms section at the bottom of the article. | Yet another way of classifying MOS is via {{w|Minkowski's question-mark function}}. Here ?(''x''), {{nowrap|? : ℝ → R}} is a continuously increasing function which has some peculiar properties, one being that it sends rational numbers to {{w|dyadic rational}}s. Hence if ''q'' is a rational number {{nowrap|0 < ''q'' < 1}} in use in the mediant system of classifying MOS scales, {{nowrap|''r'' {{=}} ?(''q'')}} = {{sfrac|''A''|2<sup>''n''</sup>}} will be a dyadic rational number which can also be used. Note that the ? function is invertible, and it and its inverse function, the Box function, have code given for them in the algorithms section at the bottom of the article. | ||
The integer ''n'' in the denominator of ''r'' (with ''A'' assumed to be odd) is the order (or {{nowrap|''n'' + 1}} is, according to some sources) of ''q'' in the {{w|Stern–Brocot tree}}. The two neighboring numbers of order {{nowrap|''n'' + 1}}, which define the range of propriety, can also be expressed in terms of the ? and Box functions as {{nowrap|Box(''r'' ± 2<sup>−''n'' − 1</sup>)}}. If ''r'' represents a MOS, the range of possible values for a generator of the MOS will be {{nowrap|Box(''r'') < ''g''}} {{nowrap|< Box(''r'' + 2<sup>−''n''</sup>)}}, and the proper generators will be {{nowrap|Box(''r'') < ''g'' < Box(''r'' + 2<sup>−''n'' − 1</sup>)}}. So, for example, the MOS denoted by 3/2048 will be between Box(3/2048) and Box(4/2048), which means that {{nowrap|{{frac|2|21}} < ''g'' < {{frac|1|10}}}}, and will be proper if {{nowrap|{{frac|2|21}} < ''g''}} < {{frac|3|31 | The integer ''n'' in the denominator of ''r'' (with ''A'' assumed to be odd) is the order (or {{nowrap|''n'' + 1}} is, according to some sources) of ''q'' in the {{w|Stern–Brocot tree}}. The two neighboring numbers of order {{nowrap|''n'' + 1}}, which define the range of propriety, can also be expressed in terms of the ? and Box functions as {{nowrap|Box(''r'' ± 2<sup>−''n'' − 1</sup>)}}. If ''r'' represents a MOS, the range of possible values for a generator of the MOS will be {{nowrap|Box(''r'') < ''g''}} {{nowrap|< Box(''r'' + 2<sup>−''n''</sup>)}}, and the proper generators will be {{nowrap|Box(''r'') < ''g'' < Box(''r'' + 2<sup>−''n'' − 1</sup>)}}. So, for example, the MOS denoted by 3/2048 will be between Box(3/2048) and Box(4/2048), which means that {{nowrap|{{frac|2|21}} < ''g'' < {{frac|1|10}}}}, and will be proper if {{nowrap|{{frac|2|21}} < ''g''}} < {{frac|3|31}}. Hence 7/72, a generator for miracle temperament, will define a MOS but it will not be proper since {{nowrap|{{frac|7|72}} > {{frac|3|31}}}} {{nowrap|{{=}} Box({{frac|3|2048}} + {{frac|1|4096}})}}. | ||
== MOS in equal tunings == | == MOS in equal tunings == | ||