Mathematics of MOS: Difference between revisions

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== MOS in equal tunings ==

In an equal tuning, all intervals are integer multiples of a smallest unit. If the equal tuning is ''N''-edo and the period is an octave, the sizes of the large and small steps will be {{~~ffac~~|''p''|''N''}} and {{frac|''q''|''N''}}, with {{nowrap|''p'' > ''q''}}. We then have {{nowrap|''L''({{frac|''p''|''N''}}) + ''s''({{frac|''q''|''N''}}) {{=}} 1}}, which on multiplying through by ''N'' gives us {{nowrap|''Lp'' + ''sq'' {{=}} N}}, which is a linear diophantine equation. Solving this by standard methods, and requiring ''L'' and ''s'' to be positive, gives us the [L, s] pair for the MOS. If some other quantity of equal steps gives the period, we may make the appropriate adjustment.

In an equal tuning, all intervals are integer multiples of a smallest unit. If the equal tuning is ''N''-edo and the period is an octave, the sizes of the large and small steps will be {{frac|''p''|''N''}} and {{frac|''q''|''N''}}, with {{nowrap|''p'' > ''q''}}. We then have {{nowrap|''L''({{frac|''p''|''N''}}) + ''s''({{frac|''q''|''N''}}) {{=}} 1}}, which on multiplying through by ''N'' gives us {{nowrap|''Lp'' + ''sq'' {{=}} N}}, which is a linear diophantine equation. Solving this by standard methods, and requiring ''L'' and ''s'' to be positive, gives us the [L, s] pair for the MOS. If some other quantity of equal steps gives the period, we may make the appropriate adjustment.

== Blackwood ''R'' constant ==

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 == MOS in equal tunings ==
-In an equal tuning, all intervals are integer multiples of a smallest unit. If the equal tuning is ''N''-edo and the period is an octave, the sizes of the large and small steps will be {{ffac|''p''|''N''}} and {{frac|''q''|''N''}}, with {{nowrap|''p'' &gt; ''q''}}. We then have {{nowrap|''L''({{frac|''p''|''N''}}) + ''s''({{frac|''q''|''N''}}) {{=}} 1}}, which on multiplying through by ''N'' gives us {{nowrap|''Lp'' + ''sq'' {{=}} N}}, which is a linear diophantine equation. Solving this by standard methods, and requiring ''L'' and ''s'' to be positive, gives us the [L,&nbsp;s] pair for the MOS. If some other quantity of equal steps gives the period, we may make the appropriate adjustment.
+In an equal tuning, all intervals are integer multiples of a smallest unit. If the equal tuning is ''N''-edo and the period is an octave, the sizes of the large and small steps will be {{frac|''p''|''N''}} and {{frac|''q''|''N''}}, with {{nowrap|''p'' &gt; ''q''}}. We then have {{nowrap|''L''({{frac|''p''|''N''}}) + ''s''({{frac|''q''|''N''}}) {{=}} 1}}, which on multiplying through by ''N'' gives us {{nowrap|''Lp'' + ''sq'' {{=}} N}}, which is a linear diophantine equation. Solving this by standard methods, and requiring ''L'' and ''s'' to be positive, gives us the [L,&nbsp;s] pair for the MOS. If some other quantity of equal steps gives the period, we may make the appropriate adjustment.
 == Blackwood ''R'' constant ==