Talk:13/10: Difference between revisions

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: Now, I'm not sure this line of reasoning is strong enough to specifically predict a <1¢ error between (13/10)^4 and 20/7, and similarly between (15/13)^4 and 567/320, so maybe the remarkable level of accuracy is a coincidence, but I think the vague relationship of approaching by tracking the relative errors has some value. And of course, as you mentioned, intervals that are sharper or flatter within their interseptimal range will accumulate more or less error, and the end result might not be accurate enough to provide satisfying equivalences, depending on the musical context and the error tolerance in that context.

: By the way, the table already shows that (13/10)^2 is sharp of 27/16 by 676/675. --[[User:Fredg999|Fredg999]] ([[User talk:Fredg999|talk]]) 05:32, 19 October 2025 (UTC)

:: By your reasoning, (13/10)^4 would be sharp of (9/7)^4=6561/2401 by roughly a semitone, with the pythagorean limma being 256/243. Therefore, (13/10)^4 would be close to 6561/2401*256/243=6912/2401. This approximation is about 13.7 cents flat of (13/10)^4, which isn't really close for an interval of this complexity. Even if the semitone were something other than 256/243, it would be hard to cancel many of the 4 factors of 7 in the denominator. By the way, the differences between 81/64, 9/7, 13/10, 21/16, and 4/3 are 64/63, 91/90, 105/104, and 64/63 in that order. The 64/63 interval is considerably wider than 91/90 and 105/104, especially for a "near-equivalence" involving an interval containing 3-5 factors of 7 in its numerator or denominator.--[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 01:35, 20 October 2025 (UTC)

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 : Now, I'm not sure this line of reasoning is strong enough to specifically predict a <1¢ error between (13/10)^4 and 20/7, and similarly between (15/13)^4 and 567/320, so maybe the remarkable level of accuracy is a coincidence, but I think the vague relationship of approaching by tracking the relative errors has some value. And of course, as you mentioned, intervals that are sharper or flatter within their interseptimal range will accumulate more or less error, and the end result might not be accurate enough to provide satisfying equivalences, depending on the musical context and the error tolerance in that context.
 : By the way, the table already shows that (13/10)^2 is sharp of 27/16 by 676/675. --[[User:Fredg999|Fredg999]] ([[User talk:Fredg999|talk]]) 05:32, 19 October 2025 (UTC)
+:: By your reasoning, (13/10)^4 would be sharp of (9/7)^4=6561/2401 by roughly a semitone, with the pythagorean limma being 256/243. Therefore, (13/10)^4 would be close to 6561/2401*256/243=6912/2401. This approximation is about 13.7 cents flat of (13/10)^4, which isn't really close for an interval of this complexity. Even if the semitone were something other than 256/243, it would be hard to cancel many of the 4 factors of 7 in the denominator. By the way, the differences between 81/64, 9/7, 13/10, 21/16, and 4/3 are 64/63, 91/90, 105/104, and 64/63 in that order. The 64/63 interval is considerably wider than 91/90 and 105/104, especially for a "near-equivalence" involving an interval containing 3-5 factors of 7 in its numerator or denominator.--[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 01:35, 20 October 2025 (UTC)