Mathematics of MOS: Difference between revisions

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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.

==Blackwood R constant==

== Blackwood ''R'' constant ==

In the context of the "recognizable diatonic" scales deriving from the Farey pair [1/2, 3/5] ~~[[Wikipedia:Easley Blackwood Jr.~~|Easley Blackwood Jr.]] defined a characterizing constant R which we may generalize to any MOS as follows. If a/b < g < c/d is a generator with the given Farey pair, take the ratio of relative errors R = (bg - a~~)/(~~c - dg). Since this is a ratio of positive numbers, it is positive. As g tends towards a/b it tends to zero, and as g goes to c/d R goes to infinity. When g ~~equals (~~a + c~~)/(~~b + d) it takes the value 1, and the range of propriety is 1/2 ≤ R ≤ 2.

In the context of the "recognizable diatonic" scales deriving from the Farey pair [{{frac|1|2}}, {{frac|3|5}}] {{w|Easley Blackwood Jr.}} defined a characterizing constant ''R'' which we may generalize to any MOS as follows: If {{nowrap|{{frac|''a''|''b''}} < ''g'' < {{frac|''c''|''d''}}}} is a generator with the given Farey pair, take the ratio of relative errors {{nowrap|''R'' {{=}} {{sfrac|''bg'' − ''a''|''c'' − ''dg''}}}}. Since this is a ratio of positive numbers, it is positive. As ''g'' tends towards {{frac|''a''|''b''}}, ''R'' tends to zero, and as ''g'' goes to {{frac|''c''|''d''}}, ''R'' goes to infinity. When {{nowrap|''g'' {{=}} {{sfrac|''a'' + ''c''|''b'' + ''d''}}}} it takes the value of 1, and the range of propriety is {{nowrap|{{frac|1|2}} ≤ ''R'' ≤ 2}}.

When R ~~is less than~~ 1, it represents the ratio in (logarithmic) size between the smaller and the larger step. When it is greater than 1, it is larger/smaller. By replacing g with 1 - g if necessary, we can reduce always to the case where R>1 (or R<1 if we prefer.)

When {{nowrap|''R'' < 1}}, it represents the ratio in (logarithmic) size between the smaller and the larger step. When it is greater than 1, it is larger/smaller. By replacing ''g'' with {{nowrap|1 − ''g''}} if necessary, we can reduce always to the case where {{nowrap|''R'' > 1}} (or {{nowrap|''R'' < 1}} if we prefer.)

==Algorithms==

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 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.
-==Blackwood R constant==
+== Blackwood ''R'' constant ==
-In the context of the "recognizable diatonic" scales deriving from the Farey pair [1/2, 3/5] [[Wikipedia:Easley Blackwood Jr.|Easley Blackwood Jr.]] defined a characterizing constant R which we may generalize to any MOS as follows. If a/b < g < c/d is a generator with the given Farey pair, take the ratio of relative errors R = (bg - a)/(c - dg). Since this is a ratio of positive numbers, it is positive. As g tends towards a/b it tends to zero, and as g goes to c/d R goes to infinity. When g equals (a + c)/(b + d) it takes the value 1, and the range of propriety is 1/2 ≤ R ≤ 2.
+In the context of the "recognizable diatonic" scales deriving from the Farey pair [{{frac|1|2}},&nbsp;{{frac|3|5}}] {{w|Easley Blackwood Jr.}} defined a characterizing constant ''R'' which we may generalize to any MOS as follows: If {{nowrap|{{frac|''a''|''b''}} &lt; ''g'' &lt; {{frac|''c''|''d''}}}} is a generator with the given Farey pair, take the ratio of relative errors {{nowrap|''R'' {{=}} {{sfrac|''bg'' &minus; ''a''|''c'' &minus; ''dg''}}}}. Since this is a ratio of positive numbers, it is positive. As ''g'' tends towards {{frac|''a''|''b''}}, ''R'' tends to zero, and as ''g'' goes to {{frac|''c''|''d''}}, ''R'' goes to infinity. When {{nowrap|''g'' {{=}} {{sfrac|''a'' + ''c''|''b'' + ''d''}}}} it takes the value of 1, and the range of propriety is {{nowrap|{{frac|1|2}} &le; ''R'' &le; 2}}.
-When R is less than 1, it represents the ratio in (logarithmic) size between the smaller and the larger step. When it is greater than 1, it is larger/smaller. By replacing g with 1 - g if necessary, we can reduce always to the case where R>1 (or R<1 if we prefer.)
+When {{nowrap|''R'' &lt; 1}}, it represents the ratio in (logarithmic) size between the smaller and the larger step. When it is greater than 1, it is larger/smaller. By replacing ''g'' with {{nowrap|1 &minus; ''g''}} if necessary, we can reduce always to the case where {{nowrap|''R'' &gt; 1}} (or {{nowrap|''R'' &lt; 1}} if we prefer.)
 ==Algorithms==