Ringer scale: Difference between revisions

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== Problem of warts ==

When trying to find a maximal odd-limit for a Ringer scale, there is a problem of a [[wikipedia:combinatorial explosion|combinatorial explosion]] if we insist on checking every possible val to try to increase the odd-limit. (Note that using a second-, third-, etc. -best mapping of a prime is called "[[wart]]ing" that prime.) This is a difficult problem to solve as it means it is unclear whether a scale is as high odd-limit as it could possibly be while maintaining the constant structure property. A potential solution to this problem is to insist that we do not use a val that uses more than one wart for a prime in order to try to keep the val as accurate and faithful to the structure of JI as possible. This makes checking all vals computationally possible. However, there are serious cases, for example [[167edo]], where the "tendency" towards sharpness or flatness of an edo is so strong that we need more than one wart for a prime in order to fit the pattern and therefore potentially achieve a higher odd-limit, so this is only really a serious solution for smaller edos, and is a partial solution for larger edos that prefers edos that do not have any "tendency". This solution works for edos as big as [[80edo]], resulting in scales like [[User:Godtone#RINGER_80|Ringer 80]], which is an important example as [[80edo]] has a strong sharp tendency for its size, to the extent that it does not map [[21/16]] or [[27/16]] consistently.

When trying to find a maximal odd-limit for a Ringer scale, there is a problem of a [[wikipedia:combinatorial explosion|combinatorial explosion]] if we insist on checking every possible val to try to increase the odd-limit. (Note that using a second-, third-, etc. -best mapping of a prime is called "[[wart]]ing" that prime.) This is a difficult problem to solve as it means it is unclear whether a scale is as high odd-limit as it could possibly be while maintaining the constant structure property. A potential solution to this problem is to insist that we do not use a val that uses more than one wart for a prime in order to try to keep the val as accurate and faithful to the structure of JI as possible. This makes checking all vals computationally possible. However, there are serious cases, for example [[167edo]], where the "tendency" towards sharpness or flatness of an edo is so strong that we need more than one wart for a prime in order to fit the pattern and therefore potentially achieve a higher odd-limit, so this is only really a serious solution for smaller edos, and is a partial solution for larger edos that prefers edos that do not have any "tendency". This solution works for edos as big as [[80edo]], resulting in scales like [[User:Godtone#RINGER_80|Ringer 80]], which is an important example as [[80edo]] has a strong sharp tendency for its size, to the extent that it does not map [[21/16]] or [[27/16]] consistently. It also tends to work well for edos that are relatively "well-tuned" in the traditional [[LCJI]]-focused [[RTT]] sense.

== Ringer scales ==

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 == Problem of warts ==
-When trying to find a maximal odd-limit for a Ringer scale, there is a problem of a [[wikipedia:combinatorial explosion|combinatorial explosion]] if we insist on checking every possible val to try to increase the odd-limit. (Note that using a second-, third-, etc. -best mapping of a prime is called "[[wart]]ing" that prime.) This is a difficult problem to solve as it means it is unclear whether a scale is as high odd-limit as it could possibly be while maintaining the constant structure property. A potential solution to this problem is to insist that we do not use a val that uses more than one wart for a prime in order to try to keep the val as accurate and faithful to the structure of JI as possible. This makes checking all vals computationally possible. However, there are serious cases, for example [[167edo]], where the "tendency" towards sharpness or flatness of an edo is so strong that we need more than one wart for a prime in order to fit the pattern and therefore potentially achieve a higher odd-limit, so this is only really a serious solution for smaller edos, and is a partial solution for larger edos that prefers edos that do not have any "tendency". This solution works for edos as big as [[80edo]], resulting in scales like [[User:Godtone#RINGER_80|Ringer 80]], which is an important example as [[80edo]] has a strong sharp tendency for its size, to the extent that it does not map [[21/16]] or [[27/16]] consistently.
+When trying to find a maximal odd-limit for a Ringer scale, there is a problem of a [[wikipedia:combinatorial explosion|combinatorial explosion]] if we insist on checking every possible val to try to increase the odd-limit. (Note that using a second-, third-, etc. -best mapping of a prime is called "[[wart]]ing" that prime.) This is a difficult problem to solve as it means it is unclear whether a scale is as high odd-limit as it could possibly be while maintaining the constant structure property. A potential solution to this problem is to insist that we do not use a val that uses more than one wart for a prime in order to try to keep the val as accurate and faithful to the structure of JI as possible. This makes checking all vals computationally possible. However, there are serious cases, for example [[167edo]], where the "tendency" towards sharpness or flatness of an edo is so strong that we need more than one wart for a prime in order to fit the pattern and therefore potentially achieve a higher odd-limit, so this is only really a serious solution for smaller edos, and is a partial solution for larger edos that prefers edos that do not have any "tendency". This solution works for edos as big as [[80edo]], resulting in scales like [[User:Godtone#RINGER_80|Ringer 80]], which is an important example as [[80edo]] has a strong sharp tendency for its size, to the extent that it does not map [[21/16]] or [[27/16]] consistently. It also tends to work well for edos that are relatively "well-tuned" in the traditional [[LCJI]]-focused [[RTT]] sense.
 == Ringer scales ==