Talk:Consistency - Revision history

Dummy index: /* q-odd-limit harmonics? prime harmonics? */ re

2023-06-09T15:50:29Z

q-odd-limit harmonics? prime harmonics?: re

← Older revision		Revision as of 15:50, 9 June 2023
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	:: I have noticed that 23edo is consistent in the no-1's 7-odd-limit ({1/1(3/3 and 5/5 and 7/7), 5/3, 3/5, 7/3, 3/7, 7/5, 5/7}). (Oh, you must have said this on 11edo.) But this is not to say that we consider 23d, it was just almost consistent and similar to 23d. <del>Direct approximation on 18edo gives 9/1 to odd-number steps, it can't be p-limit 18 nor 18b.</del> Wait, now we have to derive the mapping of up-to 7 harmonics from no-1's 7-odd-limit intervals, but can't determine uniquely... ({{val\|23 36 53 64}} or {{val\|23 37 54 65}} or {{val\|23 38 55 66}} or ...) --[[User:Dummy index\|Dummy index]] ([[User talk:Dummy index\|talk]]) 13:57, 7 June 2023 (UTC)		:: I have noticed that 23edo is consistent in the no-1's 7-odd-limit ({1/1(3/3 and 5/5 and 7/7), 5/3, 3/5, 7/3, 3/7, 7/5, 5/7}). (Oh, you must have said this on 11edo.) But this is not to say that we consider 23d, it was just almost consistent and similar to 23d. <del>Direct approximation on 18edo gives 9/1 to odd-number steps, it can't be p-limit 18 nor 18b.</del> Wait, now we have to derive the mapping of up-to 7 harmonics from no-1's 7-odd-limit intervals, but can't determine uniquely... ({{val\|23 36 53 64}} or {{val\|23 37 54 65}} or {{val\|23 38 55 66}} or ...) --[[User:Dummy index\|Dummy index]] ([[User talk:Dummy index\|talk]]) 13:57, 7 June 2023 (UTC)

			Under consideration. By the way,
			<blockquote>For instance, 12edo is consistent in the no-11's, no-13's 19-odd-limit, ...</blockquote>
			Do I need the apostrophe? --[[User:Dummy index\|Dummy index]] ([[User talk:Dummy index\|talk]]) 15:50, 9 June 2023 (UTC)

Dummy index: /* q-odd-limit harmonics? prime harmonics? */

2023-06-07T17:30:50Z

q-odd-limit harmonics? prime harmonics?

← Older revision		Revision as of 17:30, 7 June 2023
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	::: All edos are consistent in the 3-odd-limit since there's only 1, 3/2 and 4/3 to consider. 1 is by definition pure and closest. Take the closest 3/2 and that guarantees its octave complement 4/3 is closest too. The consistency is enabled by the patent val and not any other vals, so you're right that 18b is "inconsistent" tho consistency is defined for equal tunings rather than equal temperaments. [[User:FloraC\|FloraC]] ([[User talk:FloraC\|talk]]) 08:45, 30 April 2023 (UTC)		::: All edos are consistent in the 3-odd-limit since there's only 1, 3/2 and 4/3 to consider. 1 is by definition pure and closest. Take the closest 3/2 and that guarantees its octave complement 4/3 is closest too. The consistency is enabled by the patent val and not any other vals, so you're right that 18b is "inconsistent" tho consistency is defined for equal tunings rather than equal temperaments. [[User:FloraC\|FloraC]] ([[User talk:FloraC\|talk]]) 08:45, 30 April 2023 (UTC)

	:: I have noticed that 23edo is consistent in the no-1's 7-odd-limit ({1/1(3/3 and 5/5 and 7/7), 5/3, 3/5, 7/3, 3/7, 7/5, 5/7}). (Oh, you must have said this on 11edo.) But this is not to say that we consider 23d, it was just almost consistent and similar to 23d. <del>Direct approximation on 18edo gives 9/1 to odd-number steps, it can't be p-limit 18 nor 18b.</del> Wait, now we have to derive the mapping of up-to 7 harmonics from no-1's 7-odd-limit, but can't determine uniquely... ({{val\|23 36 53 64}} or {{val\|23 37 54 65}} or {{val\|23 38 55 66}} or ...) --[[User:Dummy index\|Dummy index]] ([[User talk:Dummy index\|talk]]) 13:57, 7 June 2023 (UTC)		:: I have noticed that 23edo is consistent in the no-1's 7-odd-limit ({1/1(3/3 and 5/5 and 7/7), 5/3, 3/5, 7/3, 3/7, 7/5, 5/7}). (Oh, you must have said this on 11edo.) But this is not to say that we consider 23d, it was just almost consistent and similar to 23d. <del>Direct approximation on 18edo gives 9/1 to odd-number steps, it can't be p-limit 18 nor 18b.</del> Wait, now we have to derive the mapping of up-to 7 harmonics from no-1's 7-odd-limit intervals, but can't determine uniquely... ({{val\|23 36 53 64}} or {{val\|23 37 54 65}} or {{val\|23 38 55 66}} or ...) --[[User:Dummy index\|Dummy index]] ([[User talk:Dummy index\|talk]]) 13:57, 7 June 2023 (UTC)

Dummy index: /* q-odd-limit harmonics? prime harmonics? */

2023-06-07T17:27:55Z

q-odd-limit harmonics? prime harmonics?

← Older revision		Revision as of 17:27, 7 June 2023
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	::: All edos are consistent in the 3-odd-limit since there's only 1, 3/2 and 4/3 to consider. 1 is by definition pure and closest. Take the closest 3/2 and that guarantees its octave complement 4/3 is closest too. The consistency is enabled by the patent val and not any other vals, so you're right that 18b is "inconsistent" tho consistency is defined for equal tunings rather than equal temperaments. [[User:FloraC\|FloraC]] ([[User talk:FloraC\|talk]]) 08:45, 30 April 2023 (UTC)		::: All edos are consistent in the 3-odd-limit since there's only 1, 3/2 and 4/3 to consider. 1 is by definition pure and closest. Take the closest 3/2 and that guarantees its octave complement 4/3 is closest too. The consistency is enabled by the patent val and not any other vals, so you're right that 18b is "inconsistent" tho consistency is defined for equal tunings rather than equal temperaments. [[User:FloraC\|FloraC]] ([[User talk:FloraC\|talk]]) 08:45, 30 April 2023 (UTC)

	:: I have noticed that 23edo is consistent in the no-1's 7-odd-limit ({1/1(3/3 and 5/5 and 7/7), 5/3, 3/5, 7/3, 3/7, 7/5, 5/7}). (Oh, you must have said this on 11edo.) But this is not to say that we consider 23d, it was just almost consistent ~~like a~~ 23d. Direct approximation on 18edo gives 9/1 to odd-number steps, it can't be p-limit 18 nor 18b. (~~forgot sign~~) --[[User:Dummy index\|Dummy index]] ([[User talk:Dummy index\|talk]]) 13:57, 7 June 2023 (UTC)		:: I have noticed that 23edo is consistent in the no-1's 7-odd-limit ({1/1(3/3 and 5/5 and 7/7), 5/3, 3/5, 7/3, 3/7, 7/5, 5/7}). (Oh, you must have said this on 11edo.) But this is not to say that we consider 23d, it was just almost consistent and similar to 23d. <del>Direct approximation on 18edo gives 9/1 to odd-number steps, it can't be p-limit 18 nor 18b.</del> Wait, now we have to derive the mapping of up-to 7 harmonics from no-1's 7-odd-limit, but can't determine uniquely... ({{val\|23 36 53 64}} or {{val\|23 37 54 65}} or {{val\|23 38 55 66}} or ...) --[[User:Dummy index\|Dummy index]] ([[User talk:Dummy index\|talk]]) 13:57, 7 June 2023 (UTC)

Dummy index: /* q-odd-limit harmonics? prime harmonics? */

2023-06-07T14:25:04Z

q-odd-limit harmonics? prime harmonics?

← Older revision		Revision as of 14:25, 7 June 2023
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	::: All edos are consistent in the 3-odd-limit since there's only 1, 3/2 and 4/3 to consider. 1 is by definition pure and closest. Take the closest 3/2 and that guarantees its octave complement 4/3 is closest too. The consistency is enabled by the patent val and not any other vals, so you're right that 18b is "inconsistent" tho consistency is defined for equal tunings rather than equal temperaments. [[User:FloraC\|FloraC]] ([[User talk:FloraC\|talk]]) 08:45, 30 April 2023 (UTC)		::: All edos are consistent in the 3-odd-limit since there's only 1, 3/2 and 4/3 to consider. 1 is by definition pure and closest. Take the closest 3/2 and that guarantees its octave complement 4/3 is closest too. The consistency is enabled by the patent val and not any other vals, so you're right that 18b is "inconsistent" tho consistency is defined for equal tunings rather than equal temperaments. [[User:FloraC\|FloraC]] ([[User talk:FloraC\|talk]]) 08:45, 30 April 2023 (UTC)

	:: I have noticed that 23edo is consistent in the no-1's 7-odd-limit ({1/1(3/3 and 5/5 and 7/7), 5/3, 3/5, 7/3, 3/7, 7/5, 5/7}). But this is not to say that we consider 23d, it was just almost consistent like a 23d. Direct approximation on 18edo gives 9/1 to odd-number steps, it can't be p-limit 18 nor 18b. (forgot sign) --[[User:Dummy index\|Dummy index]] ([[User talk:Dummy index\|talk]]) 13:57, 7 June 2023 (UTC)		:: I have noticed that 23edo is consistent in the no-1's 7-odd-limit ({1/1(3/3 and 5/5 and 7/7), 5/3, 3/5, 7/3, 3/7, 7/5, 5/7}). (Oh, you must have said this on 11edo.) But this is not to say that we consider 23d, it was just almost consistent like a 23d. Direct approximation on 18edo gives 9/1 to odd-number steps, it can't be p-limit 18 nor 18b. (forgot sign) --[[User:Dummy index\|Dummy index]] ([[User talk:Dummy index\|talk]]) 13:57, 7 June 2023 (UTC)

Dummy index: /* q-odd-limit harmonics? prime harmonics? */

2023-06-07T13:57:56Z

q-odd-limit harmonics? prime harmonics?

← Older revision		Revision as of 13:57, 7 June 2023
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	::: All edos are consistent in the 3-odd-limit since there's only 1, 3/2 and 4/3 to consider. 1 is by definition pure and closest. Take the closest 3/2 and that guarantees its octave complement 4/3 is closest too. The consistency is enabled by the patent val and not any other vals, so you're right that 18b is "inconsistent" tho consistency is defined for equal tunings rather than equal temperaments. [[User:FloraC\|FloraC]] ([[User talk:FloraC\|talk]]) 08:45, 30 April 2023 (UTC)		::: All edos are consistent in the 3-odd-limit since there's only 1, 3/2 and 4/3 to consider. 1 is by definition pure and closest. Take the closest 3/2 and that guarantees its octave complement 4/3 is closest too. The consistency is enabled by the patent val and not any other vals, so you're right that 18b is "inconsistent" tho consistency is defined for equal tunings rather than equal temperaments. [[User:FloraC\|FloraC]] ([[User talk:FloraC\|talk]]) 08:45, 30 April 2023 (UTC)

	:: I have noticed that 23edo is consistent in the no-1's 7-odd-limit ({1/1(3/3 and 5/5 and 7/7), 5/3, 3/5, 7/3, 3/7, 7/5, 5/7}). But this is not to say that we consider 23d, it was just almost consistent like a 23d. Direct approximation on 18edo gives 9/1 to odd-number steps, it can't be p-limit 18 nor 18b.		:: I have noticed that 23edo is consistent in the no-1's 7-odd-limit ({1/1(3/3 and 5/5 and 7/7), 5/3, 3/5, 7/3, 3/7, 7/5, 5/7}). But this is not to say that we consider 23d, it was just almost consistent like a 23d. Direct approximation on 18edo gives 9/1 to odd-number steps, it can't be p-limit 18 nor 18b. (forgot sign) --[[User:Dummy index\|Dummy index]] ([[User talk:Dummy index\|talk]]) 13:57, 7 June 2023 (UTC)

Dummy index: /* q-odd-limit harmonics? prime harmonics? */

2023-06-07T13:40:37Z

q-odd-limit harmonics? prime harmonics?

← Older revision		Revision as of 13:40, 7 June 2023
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	::: All edos are consistent in the 3-odd-limit since there's only 1, 3/2 and 4/3 to consider. 1 is by definition pure and closest. Take the closest 3/2 and that guarantees its octave complement 4/3 is closest too. The consistency is enabled by the patent val and not any other vals, so you're right that 18b is "inconsistent" tho consistency is defined for equal tunings rather than equal temperaments. [[User:FloraC\|FloraC]] ([[User talk:FloraC\|talk]]) 08:45, 30 April 2023 (UTC)		::: All edos are consistent in the 3-odd-limit since there's only 1, 3/2 and 4/3 to consider. 1 is by definition pure and closest. Take the closest 3/2 and that guarantees its octave complement 4/3 is closest too. The consistency is enabled by the patent val and not any other vals, so you're right that 18b is "inconsistent" tho consistency is defined for equal tunings rather than equal temperaments. [[User:FloraC\|FloraC]] ([[User talk:FloraC\|talk]]) 08:45, 30 April 2023 (UTC)

	:: I have noticed that 23edo is consistent in the no-1's 7-limit ({1/1(3/3 and 5/5 and 7/7), 5/3, 3/5, 7/3, 3/7, 7/5, 5/7}). But this is not to say that we consider 23d, it was just almost consistent like a 23d. Direct approximation on 18edo gives 9/1 to odd-number steps, it can't be p-limit 18 nor 18b.		:: I have noticed that 23edo is consistent in the no-1's 7-odd-limit ({1/1(3/3 and 5/5 and 7/7), 5/3, 3/5, 7/3, 3/7, 7/5, 5/7}). But this is not to say that we consider 23d, it was just almost consistent like a 23d. Direct approximation on 18edo gives 9/1 to odd-number steps, it can't be p-limit 18 nor 18b.

Dummy index: /* q-odd-limit harmonics? prime harmonics? */

2023-06-07T13:37:31Z

q-odd-limit harmonics? prime harmonics?

← Older revision		Revision as of 13:37, 7 June 2023
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	::: All edos are consistent in the 3-odd-limit since there's only 1, 3/2 and 4/3 to consider. 1 is by definition pure and closest. Take the closest 3/2 and that guarantees its octave complement 4/3 is closest too. The consistency is enabled by the patent val and not any other vals, so you're right that 18b is "inconsistent" tho consistency is defined for equal tunings rather than equal temperaments. [[User:FloraC\|FloraC]] ([[User talk:FloraC\|talk]]) 08:45, 30 April 2023 (UTC)		::: All edos are consistent in the 3-odd-limit since there's only 1, 3/2 and 4/3 to consider. 1 is by definition pure and closest. Take the closest 3/2 and that guarantees its octave complement 4/3 is closest too. The consistency is enabled by the patent val and not any other vals, so you're right that 18b is "inconsistent" tho consistency is defined for equal tunings rather than equal temperaments. [[User:FloraC\|FloraC]] ([[User talk:FloraC\|talk]]) 08:45, 30 April 2023 (UTC)

			:: I have noticed that 23edo is consistent in the no-1's 7-limit ({1/1(3/3 and 5/5 and 7/7), 5/3, 3/5, 7/3, 3/7, 7/5, 5/7}). But this is not to say that we consider 23d, it was just almost consistent like a 23d. Direct approximation on 18edo gives 9/1 to odd-number steps, it can't be p-limit 18 nor 18b.

FloraC: /* q-odd-limit harmonics? prime harmonics? */ re

2023-04-30T08:45:52Z

q-odd-limit harmonics? prime harmonics?: re

← Older revision		Revision as of 08:45, 30 April 2023
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	:: 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)		:: 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)

			::: All edos are consistent in the 3-odd-limit since there's only 1, 3/2 and 4/3 to consider. 1 is by definition pure and closest. Take the closest 3/2 and that guarantees its octave complement 4/3 is closest too. The consistency is enabled by the patent val and not any other vals, so you're right that 18b is "inconsistent" tho consistency is defined for equal tunings rather than equal temperaments. [[User:FloraC\|FloraC]] ([[User talk:FloraC\|talk]]) 08:45, 30 April 2023 (UTC)

Dummy index: /* q-odd-limit harmonics? prime harmonics? */

2023-04-30T06:54:52Z

q-odd-limit harmonics? prime harmonics?

← Older revision		Revision as of 06:54, 30 April 2023
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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)		: 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)		:: 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)

Dummy index: /* q-odd-limit harmonics? prime harmonics? */ sorry understood

2023-04-30T06:52:07Z

q-odd-limit harmonics? prime harmonics?: sorry understood

← Older revision		Revision as of 06:52, 30 April 2023
Line 79:		Line 79:

	: 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)		: 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.

			:: 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)