Mina: Difference between revisions

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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]] is uniquely [[consistent]] through to the [[27-odd-limit|27-limit]], which is not very remarkable in itself ([[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]].

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]] is uniquely [[consistent]] through to the [[27-odd-limit|27-limit]], which is not very remarkable in itself ([[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 [[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-odd-limit|27-limit]] [[tonality diamond]] are tabulated, with the sizes listed in both [[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 [[patent val]] {{val| 2460 3899 5712 6906 8510 9103 10055 10450 11128 }} for 2460edo; this will not work for [[1200edo]] and cents.

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-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]] is uniquely [[consistent]] through to the [[27-odd-limit|27-limit]], which is not very remarkable in itself ([[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]].
+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]] is uniquely [[consistent]] through to the [[27-odd-limit|27-limit]], which is not very remarkable in itself ([[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 [[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-odd-limit|27-limit]] [[tonality diamond]] are tabulated, with the sizes listed in both [[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 [[patent val]] {{val| 2460 3899 5712 6906 8510 9103 10055 10450 11128 }} for 2460edo; this will not work for [[1200edo]] and cents.