Mathematics of MOS: Difference between revisions

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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, the entropy of a particular degree (for example, a third) 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.

Given an MOS scale, the '''entropy''' of a particular degree (for example, a third) 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===

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 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, the entropy of a particular degree (for example, a third) 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.
+Given an MOS scale, the '''entropy''' of a particular degree (for example, a third) 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===