Defactoring: Difference between revisions

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when it is used as a list of 5-limit commas defining a periodicity block versus when it is used as a comma basis for a temperament, namely, that in the first case the fact that the first column has a common factor of 2 and the second column has a common factor of 3 is meaningful, i.e. the 2-enfactorment will affect one dimension of the block and the 3-enfactorment will affect a different dimension of the block, or in other words, we can say that the commas here are individually enfactored rather than the entire list being enfactored, while in the second case there is no such meaning to the individual columns' factors of 2 and 3, respectively, because it would be equivalent of any form where the product of all the column factors was 6, or in other words, all that matters is that the comma-basis as a whole is 6-enfactored here. So perhaps it would be best if, for periodicity blocks, the term "enfactored" was avoided altogether, and instead commas were described as "2-torted".</ref><ref>The explanation for "why 'torsion' in the first place?" is interesting. It comes from group theory (see: https://en.wikipedia.org/wiki/Group_(mathematics)#Uniqueness_of_identity_element). In group theory, to have torsion, a group must have an element that comes back to zero after being chained 2 or more times. The number of times before coming back to zero is called the "order" of the element, sometimes also called the "period length" or "period". When the order is greater than 1 (and less than infinity), the element is said to have torsion, or to be a torsion element, and so the group it is an identity element of is said to have torsion. See also: https://en.wikipedia.org/wiki/Order_(group_theory). Clearly we can't use period (length) because period has another firmly established meaning in xenharmonics. But we could refer to torsion as "finite order greater than one", but that's quite the mouthful while still nearly as obscure.</ref>), the term "contorsion" ~~must be banished~~ from the RTT community altogether.

when it is used as a list of 5-limit commas defining a periodicity block versus when it is used as a comma basis for a temperament, namely, that in the first case the fact that the first column has a common factor of 2 and the second column has a common factor of 3 is meaningful, i.e. the 2-enfactorment will affect one dimension of the block and the 3-enfactorment will affect a different dimension of the block, or in other words, we can say that the commas here are individually enfactored rather than the entire list being enfactored, while in the second case there is no such meaning to the individual columns' factors of 2 and 3, respectively, because it would be equivalent of any form where the product of all the column factors was 6, or in other words, all that matters is that the comma-basis as a whole is 6-enfactored here. So perhaps it would be best if, for periodicity blocks, the term "enfactored" was avoided altogether, and instead commas were described as "2-torted".</ref><ref>The explanation for "why 'torsion' in the first place?" is interesting. It comes from group theory (see: https://en.wikipedia.org/wiki/Group_(mathematics)#Uniqueness_of_identity_element). In group theory, to have torsion, a group must have an element that comes back to zero after being chained 2 or more times. The number of times before coming back to zero is called the "order" of the element, sometimes also called the "period length" or "period". When the order is greater than 1 (and less than infinity), the element is said to have torsion, or to be a torsion element, and so the group it is an identity element of is said to have torsion. See also: https://en.wikipedia.org/wiki/Order_(group_theory). Clearly we can't use period (length) because period has another firmly established meaning in xenharmonics. But we could refer to torsion as "finite order greater than one", but that's quite the mouthful while still nearly as obscure.</ref>), they feel it would be better to banish the term "contorsion" from the RTT community altogether.

# A word with the same spelling was also coined with mathematical meaning outside of RTT, in the field of differential geometry: https://en.wikipedia.org/wiki/Contorsion_tensor<ref>In this field, it does definitely represent twisting, like in a Möbius strip. Also, DG contorsion is related to DG torsion by subtraction, not duality.</ref>

# A word with the same spelling was also coined with a different mathematical meaning outside of RTT, in the field of differential geometry: https://en.wikipedia.org/wiki/Contorsion_tensor<ref>In this field, it does definitely represent twisting, like in a Möbius strip. Also, DG contorsion is related to DG torsion by subtraction, not duality.</ref>

# It is prone to spelling confusion. People commonly refer to temperaments with contorsion as "contorted". But contorted is the adjective form of a different word, contortion, with a t, not an s. The proper adjective form of contorsion would be contorsioned. Would you use "torted" instead of torsioned? Or would people prefer "torsional" and "contorsional", even though that suggests only of or pertaining to in general rather than having the effect applied.<ref>If it was meant to most strongly evoke duality with torsion, it should have been spelled "cotorsion". Naming it "contorsion" is an annoying step toward "contortion" but stopping halfway there. But this isn't a strong point, because duality with torsion was the false assumption mentioned above.</ref>

# Due to its similarity with the word "contortion", the word contorsion evokes bending, twisting, and knotting. But there is nothing bendy, twisty, or knotted about the effect it has on JI lattices or tuning space.

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-when it is used as a list of 5-limit commas defining a periodicity block versus when it is used as a comma basis for a temperament, namely, that in the first case the fact that the first column has a common factor of 2 and the second column has a common factor of 3 is meaningful, i.e. the 2-enfactorment will affect one dimension of the block and the 3-enfactorment will affect a different dimension of the block, or in other words, we can say that the commas here are individually enfactored rather than the entire list being enfactored, while in the second case there is no such meaning to the individual columns' factors of 2 and 3, respectively, because it would be equivalent of any form where the product of all the column factors was 6, or in other words, all that matters is that the comma-basis as a whole is 6-enfactored here. So perhaps it would be best if, for periodicity blocks, the term "enfactored" was avoided altogether, and instead commas were described as "2-torted".</ref><ref>The explanation for "why 'torsion' in the first place?" is interesting. It comes from group theory (see: https://en.wikipedia.org/wiki/Group_(mathematics)#Uniqueness_of_identity_element). In group theory, to have torsion, a group must have an element that comes back to zero after being chained 2 or more times. The number of times before coming back to zero is called the "order" of the element, sometimes also called the "period length" or "period". When the order is greater than 1 (and less than infinity), the element is said to have torsion, or to be a torsion element, and so the group it is an identity element of is said to have torsion. See also: https://en.wikipedia.org/wiki/Order_(group_theory). Clearly we can't use period (length) because period has another firmly established meaning in xenharmonics. But we could refer to torsion as "finite order greater than one", but that's quite the mouthful while still nearly as obscure.</ref>), the term "contorsion" must be banished from the RTT community altogether.
+when it is used as a list of 5-limit commas defining a periodicity block versus when it is used as a comma basis for a temperament, namely, that in the first case the fact that the first column has a common factor of 2 and the second column has a common factor of 3 is meaningful, i.e. the 2-enfactorment will affect one dimension of the block and the 3-enfactorment will affect a different dimension of the block, or in other words, we can say that the commas here are individually enfactored rather than the entire list being enfactored, while in the second case there is no such meaning to the individual columns' factors of 2 and 3, respectively, because it would be equivalent of any form where the product of all the column factors was 6, or in other words, all that matters is that the comma-basis as a whole is 6-enfactored here. So perhaps it would be best if, for periodicity blocks, the term "enfactored" was avoided altogether, and instead commas were described as "2-torted".</ref><ref>The explanation for "why 'torsion' in the first place?" is interesting. It comes from group theory (see: https://en.wikipedia.org/wiki/Group_(mathematics)#Uniqueness_of_identity_element). In group theory, to have torsion, a group must have an element that comes back to zero after being chained 2 or more times. The number of times before coming back to zero is called the "order" of the element, sometimes also called the "period length" or "period". When the order is greater than 1 (and less than infinity), the element is said to have torsion, or to be a torsion element, and so the group it is an identity element of is said to have torsion. See also: https://en.wikipedia.org/wiki/Order_(group_theory). Clearly we can't use period (length) because period has another firmly established meaning in xenharmonics. But we could refer to torsion as "finite order greater than one", but that's quite the mouthful while still nearly as obscure.</ref>), they feel it would be better to banish the term "contorsion" from the RTT community altogether.
-# A word with the same spelling was also coined with mathematical meaning outside of RTT, in the field of differential geometry: https://en.wikipedia.org/wiki/Contorsion_tensor<ref>In this field, it does definitely represent twisting, like in a Möbius strip. Also, DG contorsion is related to DG torsion by subtraction, not duality.</ref>
+# A word with the same spelling was also coined with a different mathematical meaning outside of RTT, in the field of differential geometry: https://en.wikipedia.org/wiki/Contorsion_tensor<ref>In this field, it does definitely represent twisting, like in a Möbius strip. Also, DG contorsion is related to DG torsion by subtraction, not duality.</ref>
 # It is prone to spelling confusion. People commonly refer to temperaments with contorsion as "contorted". But contorted is the adjective form of a different word, contortion, with a t, not an s. The proper adjective form of contorsion would be contorsioned. Would you use "torted" instead of torsioned? Or would people prefer "torsional" and "contorsional", even though that suggests only of or pertaining to in general rather than having the effect applied.<ref>If it was meant to most strongly evoke duality with torsion, it should have been spelled "cotorsion". Naming it "contorsion" is an annoying step toward "contortion" but stopping halfway there. But this isn't a strong point, because duality with torsion was the false assumption mentioned above.</ref>
 # Due to its similarity with the word "contortion", the word contorsion evokes bending, twisting, and knotting. But there is nothing bendy, twisty, or knotted about the effect it has on JI lattices or tuning space.