Mathematics of MOS: Difference between revisions

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== Classification of MOS scales ==

Since MOS scales always consist of some number of large steps and some number of small steps, they can be classified simply by the number of large steps and the number of small steps, in the form #L#s--e.g., the diatonic scale can be described as ~~5L2s~~ (5 large steps and 2 small steps) or simply [5, 2]. It is typical to ignore the period when specifying MOS scales and instead use the number of large and small steps that make up the interval of equivalence (typically assumed to be the octave--a frequency ratio of 2/1--unless otherwise specified). For instance, the diminished scale in 12-TET is typically classified as 4L4s rather than 1L1s, since there are 4 large and 4 small steps that make up an octave.

Since MOS scales always consist of some number of large steps and some number of small steps, they can be classified simply by the number of large steps and the number of small steps, in the form #L #s—e.g., the diatonic scale can be described as 5L 2s (5 large steps and 2 small steps) or simply [5, 2]. It is typical to ignore the period when specifying MOS scales and instead use the number of large and small steps that make up the interval of equivalence (typically assumed to be the octave—a frequency ratio of 2/1—unless otherwise specified). For instance, the diminished scale in 12-TET is typically classified as 4L4s rather than 1L1s, since there are 4 large and 4 small steps that make up an octave.

Alternatively, we could give a mediant for a Farey pair associated to the MOS, where this mediant is less than any generator for the MOS. In other words, we use the right hand part of the Farey pair interval, which means we must replace g with 1-g and use the complementary pair if g is in the left hand side. This method is rarely used in discussions, however.

Alternatively, we could give a mediant for a Farey pair associated to the MOS, where this mediant is less than any generator for the MOS. In other words, we use the right hand part of the Farey pair interval, which means we must replace ''g'' with {{nowrap|1 − ''g''}} and use the complementary pair if ''g'' is in the left hand side. This method is rarely used in discussions, however.

The two systems are equivalent; in the Algorithms section you will find code for routines starting from the mediant and going to the Ls pair (the "Ls" routine) and for starting from an Ls pair and going to the mediant (the "medi" routine.) The Ls routine uses ~~[[Wikipedia:Modular_multiplicative_inverse~~|modular ~~inverses]]~~, whereas the medi routine uses continued fractions.

The two systems are equivalent; in the Algorithms section you will find code for routines starting from the mediant and going to the Ls pair (the "Ls" routine) and for starting from an Ls pair and going to the mediant (the "medi" routine.) The Ls routine uses {{w|modular multiplicative inverse}}s, whereas the medi routine uses continued fractions.

If the period is assumed to be 2^(1/n) for some integer n, we can give instead the total number of large and small steps in the octave, instead of just the period, and this is commonly done. In this case, GCD(L, s) gives the number of periods in an octave.

If the period is assumed to be 2<sup>{{frac|1|''n''}}</sup> for some integer ''n'', we can give instead the total number of large and small steps in the octave, instead of just the period, and this is commonly done. In this case, GCD(L, s) gives the number of periods in an octave.

== Classification via the ? function ==

@@ Line 63: / Line 63: @@
 == Classification of MOS scales ==
-Since MOS scales always consist of some number of large steps and some number of small steps, they can be classified simply by the number of large steps and the number of small steps, in the form #L#s--e.g., the diatonic scale can be described as 5L2s (5 large steps and 2 small steps) or simply [5, 2]. It is typical to ignore the period when specifying MOS scales and instead use the number of large and small steps that make up the interval of equivalence (typically assumed to be the octave--a frequency ratio of 2/1--unless otherwise specified). For instance, the diminished scale in 12-TET is typically classified as 4L4s rather than 1L1s, since there are 4 large and 4 small steps that make up an octave.
+Since MOS scales always consist of some number of large steps and some number of small steps, they can be classified simply by the number of large steps and the number of small steps, in the form #L&nbsp;#s&mdash;e.g., the diatonic scale can be described as 5L&nbsp;2s (5 large steps and 2 small steps) or simply [5,&nbsp;2]. It is typical to ignore the period when specifying MOS scales and instead use the number of large and small steps that make up the interval of equivalence (typically assumed to be the octave&mdash;a frequency ratio of 2/1&mdash;unless otherwise specified). For instance, the diminished scale in 12-TET is typically classified as 4L4s rather than 1L1s, since there are 4 large and 4 small steps that make up an octave.
-Alternatively, we could give a mediant for a Farey pair associated to the MOS, where this mediant is less than any generator for the MOS. In other words, we use the right hand part of the Farey pair interval, which means we must replace g with 1-g and use the complementary pair if g is in the left hand side. This method is rarely used in discussions, however.
+Alternatively, we could give a mediant for a Farey pair associated to the MOS, where this mediant is less than any generator for the MOS. In other words, we use the right hand part of the Farey pair interval, which means we must replace ''g'' with {{nowrap|1 &minus; ''g''}} and use the complementary pair if ''g'' is in the left hand side. This method is rarely used in discussions, however.
-The two systems are equivalent; in the Algorithms section you will find code for routines starting from the mediant and going to the Ls pair (the "Ls" routine) and for starting from an Ls pair and going to the mediant (the "medi" routine.) The Ls routine uses [[Wikipedia:Modular_multiplicative_inverse|modular inverses]], whereas the medi routine uses continued fractions.
+The two systems are equivalent; in the Algorithms section you will find code for routines starting from the mediant and going to the Ls pair (the "Ls" routine) and for starting from an Ls pair and going to the mediant (the "medi" routine.) The Ls routine uses {{w|modular multiplicative inverse}}s, whereas the medi routine uses continued fractions.
-If the period is assumed to be 2^(1/n) for some integer n, we can give instead the total number of large and small steps in the octave, instead of just the period, and this is commonly done. In this case, GCD(L, s) gives the number of periods in an octave.
+If the period is assumed to be 2<sup>{{frac|1|''n''}}</sup> for some integer ''n'', we can give instead the total number of large and small steps in the octave, instead of just the period, and this is commonly done. In this case, GCD(L,&nbsp;s) gives the number of periods in an octave.
 == Classification via the ? function ==