Mathematics of MOS: Difference between revisions

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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 {{frac|''a''|''b''}} and {{frac|''c''|''d''}} are a [[Wikipedia:Farey sequence#Farey neighbours|Farey pair]], meaning that {{nowrap|{{frac|''a''|''b''}} < {{frac|''c''|''d''}}}} and {{nowrap|''bc'' − ''ad'' {{=}} 1}}. If {{nowrap|''g'' {{=}} (1 − ''t''){{frac|''a''|''b''}} + (''t''){{frac|''c''|''d''}}}} for {{nowrap|0 ≤ ''t'' ≤ 1}}, then when {{nowrap|''t'' {{=}} 0}}, the scale generated by ''g'' will consist of an equal division of 1 (representing P) into steps of size {{frac|1|''b''}}, and when {{nowrap|''t'' {{=}} 1}} into steps of size {{frac|1|''d''}}. In between, when {{nowrap|''t'' {{=}} {{sfrac|''b''|''b'' + ''d''}}}}, we obtain a generator equal to the [[Wikipedia:Mediant_%28mathematics%29|mediant]] {{nowrap|''m'' {{=}} {{sfrac|''a'' + ''c''|''b'' + ''d''}}}} and which will divide the period into {{nowrap|''b'' + ''d''}} equal steps. For all other values {{nowrap|{{frac|''a''|''b''}} < ''g'' < {{frac|''c''|''d''}}}} we obtain two different sizes of steps, the small steps ''s'', and the large steps ''L'', with the total number of steps {{nowrap|''b'' + ''d''}}, and these scales are the MOS associated to the Farey pair. When ''g'' is between {{frac|''a''|''b''}} and ''m'', there will be ''b'' large steps and ''d'' small steps, and when it is between ''m'' and {{frac|''c''|''d''}}, ''d'' large steps and ''b'' small ones.

Suppose the fractions {{frac|''a''|''b''}} and {{frac|''c''|''d''}} are a [[Wikipedia:Farey sequence#Farey neighbours|Farey pair]], meaning that {{nowrap|{{frac|''a''|''b''}} < {{frac|''c''|''d''}}}} and {{nowrap|''bc'' − ''ad'' {{=}} 1}}. If {{nowrap|''g'' {{=}} (1 − ''t'') {{frac|''a''|''b''}} + (''t'') {{frac|''c''|''d''}}}} for {{nowrap|0 ≤ ''t'' ≤ 1}}, then when {{nowrap|''t'' {{=}} 0}}, the scale generated by ''g'' will consist of an equal division of 1 (representing P) into steps of size {{frac|1|''b''}}, and when {{nowrap|''t'' {{=}} 1}} into steps of size {{frac|1|''d''}}. In between, when {{nowrap|''t'' {{=}} {{sfrac|''b''|''b'' + ''d''}}}}, we obtain a generator equal to the [[Wikipedia:Mediant_%28mathematics%29|mediant]] {{nowrap|''m'' {{=}} {{sfrac|''a'' + ''c''|''b'' + ''d''}}}} and which will divide the period into {{nowrap|''b'' + ''d''}} equal steps. For all other values {{nowrap|{{frac|''a''|''b''}} < ''g'' < {{frac|''c''|''d''}}}} we obtain two different sizes of steps, the small steps ''s'', and the large steps ''L'', with the total number of steps {{nowrap|''b'' + ''d''}}, and these scales are the MOS associated to the Farey pair. When ''g'' is between {{frac|''a''|''b''}} and ''m'', there will be ''b'' large steps and ''d'' small steps, and when it is between ''m'' and {{frac|''c''|''d''}}, ''d'' large steps and ''b'' small ones.

While all the scales constructed by generators ''g'' with {{nowrap|{{frac|''a''|''b''}} < ''g'' < {{frac|''c''|''d''}}}} with the exception of the mediant which gives an equal tuning are MOS, not all the scales are [[Wikipedia:Rothenberg_propriety|proper]] in the sense of Rothenberg. The ''range of propriety'' for MOS is {{nowrap|{{sfrac|2''a'' + ''c''|2''b'' + ''d''}} ≤ ''g''}} ≤ {{sfrac|''a'' + 2''c''|''b'' + 2''d''}}, where MOS coming from a Farey pair ({{frac|''a''|''b''}}, {{frac|''c''|''d''}}) are proper when in this range, and improper (unless the MOS has only one small step) when out of it. If {{nowrap|{{sfrac|2''a'' + ''c''|2''b'' + ''d''}} < ''g''}} < {{sfrac|''a'' + 2''c''|''b'' + 2''d''}}, then the scales are strictly proper. Hence, the diatonic scale in 12et, with generator 7/12, is proper but not strictly proper since starting from the pair ({{frac|1|2}}, {{frac|3|5}}) we find the range of propriety for these seven-note MOS to be [{{frac|5|9}}, {{frac|7|12}}].

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 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 {{frac|''a''|''b''}} and {{frac|''c''|''d''}} are a [[Wikipedia:Farey sequence#Farey neighbours|Farey pair]], meaning that {{nowrap|{{frac|''a''|''b''}} &lt; {{frac|''c''|''d''}}}} and {{nowrap|''bc'' &minus; ''ad'' {{=}} 1}}. If {{nowrap|''g'' {{=}} (1 &minus; ''t''){{frac|''a''|''b''}} + (''t''){{frac|''c''|''d''}}}} for {{nowrap|0 &le; ''t'' &le; 1}}, then when {{nowrap|''t'' {{=}} 0}}, the scale generated by ''g'' will consist of an equal division of 1 (representing P) into steps of size {{frac|1|''b''}}, and when {{nowrap|''t'' {{=}} 1}} into steps of size {{frac|1|''d''}}. In between, when {{nowrap|''t'' {{=}} {{sfrac|''b''|''b'' + ''d''}}}}, we obtain a generator equal to the [[Wikipedia:Mediant_%28mathematics%29|mediant]] {{nowrap|''m'' {{=}} {{sfrac|''a'' + ''c''|''b'' + ''d''}}}} and which will divide the period into {{nowrap|''b'' + ''d''}} equal steps. For all other values {{nowrap|{{frac|''a''|''b''}} &lt; ''g'' &lt; {{frac|''c''|''d''}}}} we obtain two different sizes of steps, the small steps ''s'', and the large steps ''L'', with the total number of steps {{nowrap|''b'' + ''d''}}, and these scales are the MOS associated to the Farey pair. When ''g'' is between {{frac|''a''|''b''}} and ''m'', there will be ''b'' large steps and ''d'' small steps, and when it is between ''m'' and {{frac|''c''|''d''}}, ''d'' large steps and ''b'' small ones.
+Suppose the fractions {{frac|''a''|''b''}} and {{frac|''c''|''d''}} are a [[Wikipedia:Farey sequence#Farey neighbours|Farey pair]], meaning that {{nowrap|{{frac|''a''|''b''}} &lt; {{frac|''c''|''d''}}}} and {{nowrap|''bc'' &minus; ''ad'' {{=}} 1}}. If {{nowrap|''g'' {{=}} (1 &minus; ''t'')&#x200A;{{frac|''a''|''b''}} + (''t'')&#x200A;{{frac|''c''|''d''}}}} for {{nowrap|0 &le; ''t'' &le; 1}}, then when {{nowrap|''t'' {{=}} 0}}, the scale generated by ''g'' will consist of an equal division of 1 (representing P) into steps of size {{frac|1|''b''}}, and when {{nowrap|''t'' {{=}} 1}} into steps of size {{frac|1|''d''}}. In between, when {{nowrap|''t'' {{=}} {{sfrac|''b''|''b'' + ''d''}}}}, we obtain a generator equal to the [[Wikipedia:Mediant_%28mathematics%29|mediant]] {{nowrap|''m'' {{=}} {{sfrac|''a'' + ''c''|''b'' + ''d''}}}} and which will divide the period into {{nowrap|''b'' + ''d''}} equal steps. For all other values {{nowrap|{{frac|''a''|''b''}} &lt; ''g'' &lt; {{frac|''c''|''d''}}}} we obtain two different sizes of steps, the small steps ''s'', and the large steps ''L'', with the total number of steps {{nowrap|''b'' + ''d''}}, and these scales are the MOS associated to the Farey pair. When ''g'' is between {{frac|''a''|''b''}} and ''m'', there will be ''b'' large steps and ''d'' small steps, and when it is between ''m'' and {{frac|''c''|''d''}}, ''d'' large steps and ''b'' small ones.
 While all the scales constructed by generators ''g'' with {{nowrap|{{frac|''a''|''b''}} &lt; ''g'' &lt; {{frac|''c''|''d''}}}} with the exception of the mediant which gives an equal tuning are MOS, not all the scales are [[Wikipedia:Rothenberg_propriety|proper]] in the sense of Rothenberg. The ''range of propriety'' for MOS is {{nowrap|{{sfrac|2''a'' + ''c''|2''b'' + ''d''}} &le; ''g''}} &le;&nbsp;{{sfrac|''a'' + 2''c''|''b'' + 2''d''}}, where MOS coming from a Farey pair ({{frac|''a''|''b''}}, {{frac|''c''|''d''}}) are proper when in this range, and improper (unless the MOS has only one small step) when out of it. If {{nowrap|{{sfrac|2''a'' + ''c''|2''b'' + ''d''}} &lt; ''g''}} &lt;&nbsp;{{sfrac|''a'' + 2''c''|''b'' + 2''d''}}, then the scales are strictly proper. Hence, the diatonic scale in 12et, with generator 7/12, is proper but not strictly proper since starting from the pair ({{frac|1|2}},&nbsp;{{frac|3|5}}) we find the range of propriety for these seven-note MOS to be [{{frac|5|9}},&nbsp;{{frac|7|12}}].