Mathematics of MOS: Difference between revisions

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

Given a generator g, we can find MOS for g with period 1 by means of the [[Wikipedia:Continued_fraction#Semiconvergents|semiconvergents]] to g. A pair of successive semiconvergents have the property that they define a Farey pair, and when g is contained in the pair, that is, a/b < g < c/d, we have defined a MOS for g with b+d as the number of notes in the MOS, with b notes of one size and d of the other.

Given a generator g, we can find an MOS for g with period 1 by means of the [[Wikipedia:Continued_fraction#Semiconvergents|semiconvergents]] to g. A pair of successive semiconvergents have the property that they define a Farey pair, and when g is contained in the pair, that is, a/b < g < c/d, we have defined a MOS for g with b+d as the number of notes in the MOS, with b notes of one size and d of the other.

For example, suppose we want MOS for 1/4-comma meantone. The generator will then be log2(5)/4, which has semiconvergents 1/2, 2/3, 3/5, 4/7, 7/12, 11/19, 18/31, 29/50, 47/81, 65/112... If we settle on 31 as a good size for our MOS, we see 18/31 is the mediant between the Farey pair 11/19 and 7/12, for which the range of strict propriety is 29/50 < x < 25/43. Since g is in that range and not equal to 18/31, we will get a strictly proper MOS.

For example, suppose we want an MOS for 1/4-comma meantone. The generator will then be log2(5)/4, which has semiconvergents 1/2, 2/3, 3/5, 4/7, 7/12, 11/19, 18/31, 29/50, 47/81, 65/112... If we settle on 31 as a good size for our MOS, we see 18/31 is the mediant between the Farey pair 11/19 and 7/12, for which the range of strict propriety is 29/50 < x < 25/43. Since g is in that range and not equal to 18/31, we will get a strictly proper MOS.

==Entropy of a scale degree==

Given an MOS scale (for example, a third) of the scale, the entropy of a particular degree is how informative the degree is of what mode the MOS scale is in (assuming all modes are equally likely). It quantifies the intuition that in the diatonic scale thirds and sixths are the most "informative" of the diatonic mode, and fourths and fifths are the least informative. (The total information that we need for a modal rotation is log2(N/p) bits, where N is the size of the MOS and p is the number of periods per octave. The larger the scale is, the more tones we need to hear to determine what the mode is.) It is simply the Shannon entropy for the major/minor versions for the interval class: If N is the size of the MOS and the interval class in question is represented by a rank-2 interval mg (mod 1200 cents), then the class's entropy is log2(m/N)*m/N + log2((N-m)/N)*(N-m)/N.

===5 notes===

===6 notes===

===7 notes===

===8 notes===

===9 notes===

===10 notes===

=Visualizing MOS: Generator Chains, Pitch Space, and Hierarchies=

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 While all the scales constructed by generators g with a/b < g < c/d with the exception of the mediant which gives an equal temperament are MOS, not all the scales are [[Wikipedia:Rothenberg_propriety|proper]] in the sense of Rothenberg. The ''range of propriety'' for MOS is (2a + c)/(2b + d) ≤ g ≤ (a + 2c)/(b + 2d), where MOS coming from a Farey pair (a/b, c/d) are proper when in this range, and improper (unless the MOS has only one small step) when out of it. If (2a + c)/(2b + d) < g < (a + 2c)/(b + 2d), 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 (1/2, 3/5) we find the range of propriety for these seven-note MOS to be [5/9, 7/12].
-Given a generator g, we can find MOS for g with period 1 by means of the [[Wikipedia:Continued_fraction#Semiconvergents|semiconvergents]] to g. A pair of successive semiconvergents have the property that they define a Farey pair, and when g is contained in the pair, that is, a/b < g < c/d, we have defined a MOS for g with b+d as the number of notes in the MOS, with b notes of one size and d of the other.
+Given a generator g, we can find an MOS for g with period 1 by means of the [[Wikipedia:Continued_fraction#Semiconvergents|semiconvergents]] to g. A pair of successive semiconvergents have the property that they define a Farey pair, and when g is contained in the pair, that is, a/b < g < c/d, we have defined a MOS for g with b+d as the number of notes in the MOS, with b notes of one size and d of the other.
-For example, suppose we want MOS for 1/4-comma meantone. The generator will then be log2(5)/4, which has semiconvergents 1/2, 2/3, 3/5, 4/7, 7/12, 11/19, 18/31, 29/50, 47/81, 65/112... If we settle on 31 as a good size for our MOS, we see 18/31 is the mediant between the Farey pair 11/19 and 7/12, for which the range of strict propriety is 29/50 < x < 25/43. Since g is in that range and not equal to 18/31, we will get a strictly proper MOS.
+For example, suppose we want an MOS for 1/4-comma meantone. The generator will then be log2(5)/4, which has semiconvergents 1/2, 2/3, 3/5, 4/7, 7/12, 11/19, 18/31, 29/50, 47/81, 65/112... If we settle on 31 as a good size for our MOS, we see 18/31 is the mediant between the Farey pair 11/19 and 7/12, for which the range of strict propriety is 29/50 < x < 25/43. Since g is in that range and not equal to 18/31, we will get a strictly proper MOS.
+==Entropy of a scale degree==
+Given an MOS scale (for example, a third) of the scale, the entropy of a particular degree is how informative the degree is of what mode the MOS scale is in (assuming all modes are equally likely). It quantifies the intuition that in the diatonic scale thirds and sixths are the most "informative" of the diatonic mode, and fourths and fifths are the least informative. (The total information that we need for a modal rotation is log<sub>2</sub>(N/p) bits, where N is the size of the MOS and p is the number of periods per octave. The larger the scale is, the more tones we need to hear to determine what the mode is.) It is simply the Shannon entropy for the major/minor versions for the interval class: If N is the size of the MOS and the interval class in question is represented by a rank-2 interval mg (mod 1200 cents), then the class's entropy is log<sub>2</sub>(m/N)*m/N + log<sub>2</sub>((N-m)/N)*(N-m)/N.
+===5 notes===
+===6 notes===
+===7 notes===
+===8 notes===
+===9 notes===
+===10 notes===
 =Visualizing MOS: Generator Chains, Pitch Space, and Hierarchies=