Mina: Difference between revisions

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The ''mina'' is a unit of interval size which has been proposed by [[George_Secor|George Secor]] and [[Dave_Keenan|Dave Keenan]], and which is defined as 1/2460 of an [[Octave|octave]], the step size of [[2460edo|2460edo]]. 2460 is divisible by both 12 and 41, two important systems, and it's been suggested that degrees and minutes can be used to express values in it, so that for instance 3/2, which is 1439 minas, could be denoted by 23°59', meaning very slightly flat of the 24\41 [[41edo|41edo]] fifths. This works out since 41 * 60 = 2460; an octave is therefore expressed as if it were an angle of 41 degrees.

Other popular systems that can be represented exactly in whole numbers of minas include [[10edo|10edo]] and [[15edo|15edo]]. Moreover a cent is exactly 2.05 ~~[[mina|mina]]s~~, and a mem, 1\205 octaves, is exactly 12 minas.

Other popular systems that can be represented exactly in whole numbers of minas include [[10edo|10edo]] and [[15edo|15edo]]. Moreover a cent is exactly 2.05 minas, and a ''mem'', 1\205 octaves, is exactly 12 minas.

The following table lists some intervals which may be represented exactly in minas and in degrees and minutes, with the sizes listed in both [[cent|cent]]s and minas and expressed as degrees and minutes.

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Another notable feature of the mina is the accuracy and breadth of ~~it's approximation~~ to just intervals. Accordingly it is hardly necessary to express intervals in non-integer values of mina, something that arguably cannot be said of cents. [[2460edo|2460edo]] It is uniquely [[consistent|consistent]] through to the [[27-limit|27-limit]], which is not very remarkable in itself ([[388edo|388edo]] is the first such system), but what is remarkable is the degree of accuracy to which it represents the 27-limit intervals. It is also a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]] and has a lower 19-limit [[Tenney-Euclidean_temperament_measures#TE simple badness|relative error]] than any edo until [[3395edo|3395]], and a lower 23-limit relative error than any until [[8269edo|8269]]. Also it has a lower 23-limit [[Tenney-Euclidean_metrics#Logflat TE badness| TE ~~loglfat~~ badness]] than any smaller edo and less than any until [[16808edo|16808]].

Another notable feature of the mina is the accuracy and breadth of its approximations to just intervals. Accordingly it is hardly necessary to express intervals in non-integer values of mina, something that arguably cannot be said of cents. [[2460edo|2460edo]] is uniquely [[consistent|consistent]] through to the [[27-limit|27-limit]], which is not very remarkable in itself ([[388edo|388edo]] is the first such system), but what is remarkable is the degree of accuracy to which it represents the 27-limit intervals. It is also a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]] and has a lower 19-limit [[Tenney-Euclidean_temperament_measures#TE simple badness|relative error]] than any edo until [[3395edo|3395]], and a lower 23-limit relative error than any until [[8269edo|8269]]. Also it has a lower 23-limit [[Tenney-Euclidean_metrics#Logflat TE badness| TE logflat badness]] than any smaller edo and less than any until [[16808edo|16808]].

Below the intervals of the [[27-limit|27-limit]] [[Tonality_diamond|tonality diamond]] are tabulated, with the sizes listed in both [[cent|cent]]s and minas and expressed as degrees and minutes (rounded to the nearest minute). The value in minas, rounded to the nearest integer, can be found by applying the [[23-limit|23-limit]] [[Patent_val|patent val]] <2460 3899 5712 6906 8510 9103 10055 10450 11128| for 2460edo; this will not work for [[1200edo|1200edo]] and cents.

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[[Category:measure]]

[[Category:mina]]

~~[[Category:todo:unify_precision]]~~

[[Category:unit]]

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 The ''mina'' is a unit of interval size which has been proposed by [[George_Secor|George Secor]] and [[Dave_Keenan|Dave Keenan]], and which is defined as 1/2460 of an [[Octave|octave]], the step size of [[2460edo|2460edo]]. 2460 is divisible by both 12 and 41, two important systems, and it's been suggested that degrees and minutes can be used to express values in it, so that for instance 3/2, which is 1439 minas, could be denoted by 23°59', meaning very slightly flat of the 24\41 [[41edo|41edo]] fifths. This works out since 41 * 60 = 2460; an octave is therefore expressed as if it were an angle of 41 degrees.
-Other popular systems that can be represented exactly in whole numbers of minas include [[10edo|10edo]] and [[15edo|15edo]]. Moreover a cent is exactly 2.05 [[mina|mina]]s, and a mem, 1\205 octaves, is exactly 12 minas.
+Other popular systems that can be represented exactly in whole numbers of minas include [[10edo|10edo]] and [[15edo|15edo]]. Moreover a cent is exactly 2.05 minas, and a ''mem'', 1\205 octaves, is exactly 12 minas.
 The following table lists some intervals which may be represented exactly in minas and in degrees and minutes, with the sizes listed in both [[cent|cent]]s and minas and expressed as degrees and minutes.
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 |}
-Another notable feature of the mina is the accuracy and breadth of it's approximation to just intervals. Accordingly it is hardly necessary to express intervals in non-integer values of mina, something that arguably cannot be said of cents. [[2460edo|2460edo]] It is uniquely [[consistent|consistent]] through to the [[27-limit|27-limit]], which is not very remarkable in itself ([[388edo|388edo]] is the first such system), but what is remarkable is the degree of accuracy to which it represents the 27-limit intervals. It is also a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]] and has a lower 19-limit [[Tenney-Euclidean_temperament_measures#TE simple badness|relative error]] than any edo until [[3395edo|3395]], and a lower 23-limit relative error than any until [[8269edo|8269]]. Also it has a lower 23-limit [[Tenney-Euclidean_metrics#Logflat TE badness| TE loglfat badness]] than any smaller edo and less than any until [[16808edo|16808]].
+Another notable feature of the mina is the accuracy and breadth of its approximations to just intervals. Accordingly it is hardly necessary to express intervals in non-integer values of mina, something that arguably cannot be said of cents. [[2460edo|2460edo]] is uniquely [[consistent|consistent]] through to the [[27-limit|27-limit]], which is not very remarkable in itself ([[388edo|388edo]] is the first such system), but what is remarkable is the degree of accuracy to which it represents the 27-limit intervals. It is also a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]] and has a lower 19-limit [[Tenney-Euclidean_temperament_measures#TE simple badness|relative error]] than any edo until [[3395edo|3395]], and a lower 23-limit relative error than any until [[8269edo|8269]]. Also it has a lower 23-limit [[Tenney-Euclidean_metrics#Logflat TE badness| TE logflat badness]] than any smaller edo and less than any until [[16808edo|16808]].
 Below the intervals of the [[27-limit|27-limit]] [[Tonality_diamond|tonality diamond]] are tabulated, with the sizes listed in both [[cent|cent]]s and minas and expressed as degrees and minutes (rounded to the nearest minute). The value in minas, rounded to the nearest integer, can be found by applying the [[23-limit|23-limit]] [[Patent_val|patent val]] &lt;2460 3899 5712 6906 8510 9103 10055 10450 11128| for 2460edo; this will not work for [[1200edo|1200edo]] and cents.
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 [[Category:measure]]
 [[Category:mina]]
-[[Category:todo:unify_precision]]
 [[Category:unit]]