Talk:159edo: Difference between revisions

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::::::: Actually, if you think about it, the generally accepted detuning of the major third in [[12edo]] still follows the same rules that I'm laying down, as the Syntonic comma ([[81/80]]), which is responsible for that detuning, is smaller than half a step in 12edo, and it's still smaller than half a step in [[24edo]]. In fact, the [[Pythagorean comma]] is also less than half of a step in 24edo, and thus, the 3-prime and the 5-prime can both be regarded as having "complete consistency" in 24edo as well as in 12edo. However, when you start looking at [[36edo]], [[48edo]] and [[72edo]], suddenly, things don't turn out as good on this front, as the relative error percentage in these EDOs- especially for the Pythagorean comma- exceeds 50%. This is why I moved on from the larger 12-based EDOs and was finally open to detwelvulating. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 21:41, 17 January 2021 (UTC)

::::::: If you apply this same type of thinking to smaller EDOs, you see that 2edo is the first to have complete consistency in the 3-limit, but 3edo and 4edo both fail this test as the commas produced by their respective circles of fifths are larger than half of their respective step size. After that, the next EDO to have complete consistency in the 3-limit is 5edo, which accomplishes a completely consistent representation of the 3-prime as 256/243, the interval produced from a single circle of fifths in 5edo, is smaller than half of a step in 5edo. After that, the next EDO to have complete consistency in the 3-~~limit~~ is 12edo itself, as 6edo, 7edo, 8edo, 9edo, 10edo, and 11edo all fail the test- of course, as I said, 24edo, which is related to 12edo, also passes this test. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 22:50, 17 January 2021 (UTC)

::::::: If you apply this same type of thinking to smaller EDOs, you see that 2edo is the first to have complete consistency in the 3-limit, but 3edo and 4edo both fail this test as the commas produced by their respective circles of fifths are larger than half of their respective step size. After that, the next EDO to have complete consistency in the 3-limit is 5edo, which accomplishes a completely consistent representation of the 3-prime as 256/243, the interval produced from a single circle of fifths in 5edo, is smaller than half of a step in 5edo. After that, the next EDO to have complete consistency in the 3-prime is 12edo itself, as 6edo, 7edo, 8edo, 9edo, 10edo, and 11edo all fail the test- of course, as I said, 24edo, which is related to 12edo, also passes this test. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 22:50, 17 January 2021 (UTC)

::::::: So, what about the EDOs between 12edo and 24edo? Well, according to my calculations, literally none of the EDOs from 13edo to 23edo demonstrate complete consistency in the 3-limit. Even the well known [[22edo]] fails this test- looks like I've found one of that EDO's significant weaknesses, and a good enough reason for me not to use it. Anyhow, I'll continue my calculations to see what other EDOs demonstrate the kind of complete 3-prime consistency, and I'll let y'all know about the first dozen or so members of the sequence that emerges from this. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 23:12, 17 January 2021 (UTC)

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 ::::::: Actually, if you think about it, the generally accepted detuning of the major third in [[12edo]] still follows the same rules that I'm laying down, as the Syntonic comma ([[81/80]]), which is responsible for that detuning, is smaller than half a step in 12edo, and it's still smaller than half a step in [[24edo]].  In fact, the [[Pythagorean comma]] is also less than half of a step in 24edo, and thus, the 3-prime and the 5-prime can both be regarded as having "complete consistency" in 24edo as well as in 12edo.  However, when you start looking at [[36edo]], [[48edo]] and [[72edo]], suddenly, things don't turn out as good on this front, as the relative error percentage in these EDOs- especially for the Pythagorean comma- exceeds 50%.  This is why I moved on from the larger 12-based EDOs and was finally open to detwelvulating. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 21:41, 17 January 2021 (UTC)
-::::::: If you apply this same type of thinking to smaller EDOs, you see that 2edo is the first to have complete consistency in the 3-limit, but 3edo and 4edo both fail this test as the commas produced by their respective circles of fifths are larger than half of their respective step size.  After that, the next EDO to have complete consistency in the 3-limit is 5edo, which accomplishes a completely consistent representation of the 3-prime as 256/243, the interval produced from a single circle of fifths in 5edo, is smaller than half of a step in 5edo.  After that, the next EDO to have complete consistency in the 3-limit is 12edo itself, as 6edo, 7edo, 8edo, 9edo, 10edo, and 11edo all fail the test- of course, as I said, 24edo, which is related to 12edo, also passes this test. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 22:50, 17 January 2021 (UTC)
+::::::: If you apply this same type of thinking to smaller EDOs, you see that 2edo is the first to have complete consistency in the 3-limit, but 3edo and 4edo both fail this test as the commas produced by their respective circles of fifths are larger than half of their respective step size.  After that, the next EDO to have complete consistency in the 3-limit is 5edo, which accomplishes a completely consistent representation of the 3-prime as 256/243, the interval produced from a single circle of fifths in 5edo, is smaller than half of a step in 5edo.  After that, the next EDO to have complete consistency in the 3-prime is 12edo itself, as 6edo, 7edo, 8edo, 9edo, 10edo, and 11edo all fail the test- of course, as I said, 24edo, which is related to 12edo, also passes this test. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 22:50, 17 January 2021 (UTC)
+::::::: So, what about the EDOs between 12edo and 24edo?  Well, according to my calculations, literally none of the EDOs from 13edo to 23edo demonstrate complete consistency in the 3-limit.  Even the well known [[22edo]] fails this test- looks like I've found one of that EDO's significant weaknesses, and a good enough reason for me not to use it.  Anyhow, I'll continue my calculations to see what other EDOs demonstrate the kind of complete 3-prime consistency, and I'll let y'all know about the first dozen or so members of the sequence that emerges from this. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 23:12, 17 January 2021 (UTC)