Mathematics of MOS: Difference between revisions

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Let us represent the period as 1. This would be the logarithm base 2 of 2 if the period is an octave, or in general we can measure intervals by the log base P when P is the period. Suppose the fractions a/b and c/d are a [[Wikipedia:Farey_sequence#Farey_neighbours|Farey pair]], meaning that a/b < c/d and bc - ad = 1. If g = (1-t)(a/b) + t(c/d) for 0 ? t ? 1, then when t = 0, the scale generated by g will consist of an equal division of 1 (representing P) into steps of size 1/b, and when t = 1 into steps of size 1/d. In between, when t = b/(b + d), we obtain a generator equal to the [[Wikipedia:Mediant_%28mathematics%29|mediant]] (a + c)/(b + d) and which will divide the period into b+d equal steps. For all other values a/b < g < c/d we obtain two different sizes of steps, the small steps s, and the large steps L, with the total number of steps b+d, and these scales are the MOS associated to the Farey pair. When g is between a/b and (a + c)/(b + d) there will be b large steps and d small steps, and when it is between (a + c)/(b + d) and c/d, d large steps and b small ones.

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].

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.

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==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 [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.

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.)

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 Let us represent the period as 1. This would be the logarithm base 2 of 2 if the period is an octave, or in general we can measure intervals by the log base P when P is the period. Suppose the fractions a/b and c/d are a [[Wikipedia:Farey_sequence#Farey_neighbours|Farey pair]], meaning that a/b < c/d and bc - ad = 1. If g = (1-t)(a/b) + t(c/d) for 0 ? t ? 1, then when t = 0, the scale generated by g will consist of an equal division of 1 (representing P) into steps of size 1/b, and when t = 1 into steps of size 1/d. In between, when t = b/(b + d), we obtain a generator equal to the [[Wikipedia:Mediant_%28mathematics%29|mediant]] (a + c)/(b + d) and which will divide the period into b+d equal steps. For all other values a/b < g < c/d we obtain two different sizes of steps, the small steps s, and the large steps L, with the total number of steps b+d, and these scales are the MOS associated to the Farey pair. When g is between a/b and (a + c)/(b + d) there will be b large steps and d small steps, and when it is between (a + c)/(b + d) and c/d, d large steps and b small ones.
-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].
+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.
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 ==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 [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.
 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.)