Talk:Consistency: Difference between revisions

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: If I understand correctly, your concern is that consistency is defined on edos assuming an infinitely long map (val) based specifically on all prime harmonics, therefore excluding maps with composite or fractional subgroups. It is good to keep in mind that the choice of the map affects the consistency; for example, 18edo is consistent in the 7-odd-limit, but 18b (second-best mapping for prime 3) is only consistent in the 3-odd-limit, because then 5/3 wouldn't be mapped to its direct approximation. And so we find that a dual-fifth system such as 35edo has a well-defined consistency if we assume its [[integer uniform map]] ([[patent val]]) — in this case 7-odd-limit, because 9/1 is not mapped to its direct approximation —, but if we coupled 35edo with a 2.9.5.7.11 subgroup map, we would indeed have no way to reach the interval 3/1 at all. Since the goal of consistency is to check if a given equal temperament (edo + map) can map all intervals in an odd-limit to their direct approximation, it wouldn't make sense to allow subgroups that skip over certain intervals. Therefore, a uniform map, or at least a map based on a full prime limit, should be used to evaluate an equal temperament's consistency. It would be interesting to check which equal temperaments have a higher consistency limit with a "warted" map compared to the standard one; for example, 11edo is only consistent in the 3-odd-limit, but 11b is consistent in the 7-odd-limit (but not 9-odd-limit because 9/5 is not mapped to its direct approximation). --[[User:Fredg999|Fredg999]] ([[User talk:Fredg999|talk]]) 16:20, 29 April 2023 (UTC)

:: Sorry, I don't talk about we should/shoudn't use subgroups. I talk about the Opening clause of the article. The mathematical definition section is fine (but we should add the wording that determines the range of the r, odd-limit and subgroup or any other). Opening clause says... Oh, sorry I see it's no problem. 18edo in the 9-odd-limit maps 3/1 to 29 steps, 9/1 to 57 steps, therefore difference between 9/1 and 3/1 doesn't give closest approximation of 3/1.

:: Sorry, I don't talk about we should/shoudn't use subgroups. I talk about the Opening clause of the article. The mathematical definition section is fine <del>(but we should add the wording that determines the range of the r, odd-limit and subgroup or any other)</del>. Opening clause says... Oh, sorry I see it's no problem. 18edo in the 9-odd-limit maps 3/1 to 29 steps, 9/1 to 57 steps, therefore difference between 9/1 and 3/1 doesn't give closest approximation of 3/1.

:: Hence, I think 18b doesn't map 3/1 as closest approximation. Difference between 3/1 and 1/1 doesn't give closest approximation of 3/1. Wouldn't the premise of the discussion be different? --[[User:Dummy index|Dummy index]] ([[User talk:Dummy index|talk]]) 06:52, 30 April 2023 (UTC)

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 : If I understand correctly, your concern is that consistency is defined on edos assuming an infinitely long map (val) based specifically on all prime harmonics, therefore excluding maps with composite or fractional subgroups. It is good to keep in mind that the choice of the map affects the consistency; for example, 18edo is consistent in the 7-odd-limit, but 18b (second-best mapping for prime 3) is only consistent in the 3-odd-limit, because then 5/3 wouldn't be mapped to its direct approximation. And so we find that a dual-fifth system such as 35edo has a well-defined consistency if we assume its [[integer uniform map]] ([[patent val]]) — in this case 7-odd-limit, because 9/1 is not mapped to its direct approximation —, but if we coupled 35edo with a 2.9.5.7.11 subgroup map, we would indeed have no way to reach the interval 3/1 at all. Since the goal of consistency is to check if a given equal temperament (edo + map) can map all intervals in an odd-limit to their direct approximation, it wouldn't make sense to allow subgroups that skip over certain intervals. Therefore, a uniform map, or at least a map based on a full prime limit, should be used to evaluate an equal temperament's consistency. It would be interesting to check which equal temperaments have a higher consistency limit with a "warted" map compared to the standard one; for example, 11edo is only consistent in the 3-odd-limit, but 11b is consistent in the 7-odd-limit (but not 9-odd-limit because 9/5 is not mapped to its direct approximation). --[[User:Fredg999|Fredg999]] ([[User talk:Fredg999|talk]]) 16:20, 29 April 2023 (UTC)
-:: Sorry, I don't talk about we should/shoudn't use subgroups. I talk about the Opening clause of the article. The mathematical definition section is fine (but we should add the wording that determines the range of the r, odd-limit and subgroup or any other). Opening clause says... Oh, sorry I see it's no problem. 18edo in the 9-odd-limit maps 3/1 to 29 steps, 9/1 to 57 steps, therefore difference between 9/1 and 3/1 doesn't give closest approximation of 3/1.
+:: Sorry, I don't talk about we should/shoudn't use subgroups. I talk about the Opening clause of the article. The mathematical definition section is fine <del>(but we should add the wording that determines the range of the r, odd-limit and subgroup or any other)</del>. Opening clause says... Oh, sorry I see it's no problem. 18edo in the 9-odd-limit maps 3/1 to 29 steps, 9/1 to 57 steps, therefore difference between 9/1 and 3/1 doesn't give closest approximation of 3/1.
 :: Hence, I think 18b doesn't map 3/1 as closest approximation. Difference between 3/1 and 1/1 doesn't give closest approximation of 3/1. Wouldn't the premise of the discussion be different? --[[User:Dummy index|Dummy index]] ([[User talk:Dummy index|talk]]) 06:52, 30 April 2023 (UTC)