Talk:13/10: Difference between revisions

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"Because 13/10 is an interseptimal interval, stacking it four times will result in a good approximation of a septimal interval." Where does this come from? Stacking [[15/13]] four times sort of gives an approximation to [[7/4]], but it is over 22 cents sharp, so not really. From (13/10)^4 being close to [[10/7]], we know that (15/13)^4 is close to 567/320, but that interval is quite complex. It seems even more arbitrary when we consider that 13/10, 15/13, and their octave complements aren't the only interseptimal intervals; for example, ([[22/19]])^4 is close to [[9/5]], but not to any simple septimal interval. A less arbitrary fact would be two septimal intervals stacking near a pythagorean interval (e.g. 4/3, 27/16, or 256/243), but I don't think that general fact is worth including on this page either. However, it may be worth including that (13/10)^2 is sharp of 27/16 by just [[676/675]].--[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 03:34, 19 October 2025 (UTC)

: 567/320 looks complex because it has large numbers, but really it's just 7/5 above 81/64. Granted, it's more complex than 20/7 (which is 10/7 above 2/1), but it's certainly not an extremely complex septimal interval either.

: The intuition behind this property is that common interseptimal intervals are rather well centered between the two septimal intervals that serve as boundaries for the region, and likewise that common septimal intervals are rather well centered between their nearest interseptimal region and their nearest diatonic category. For example, the M3-P4 spectrum includes 81/64 (Pythagorean), then 9/7 (septimal), then 13/10 (interseptimal), then 21/16 (septimal), then 4/3 (Pythagorean). Thus an interseptimal interval has roughly 1/4 of the distance between two diatonic categories of an error to its closest septimal interval. When brought to the fourth power, the error approaches a semitone, and therefore it ends up approximating another septimal interval.

: 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)

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 "Because 13/10 is an interseptimal interval, stacking it four times will result in a good approximation of a septimal interval." Where does this come from? Stacking [[15/13]] four times sort of gives an approximation to [[7/4]], but it is over 22 cents sharp, so not really. From (13/10)^4 being close to [[10/7]], we know that (15/13)^4 is close to 567/320, but that interval is quite complex. It seems even more arbitrary when we consider that 13/10, 15/13, and their octave complements aren't the only interseptimal intervals; for example, ([[22/19]])^4 is close to [[9/5]], but not to any simple septimal interval. A less arbitrary fact would be two septimal intervals stacking near a pythagorean interval (e.g. 4/3, 27/16, or 256/243), but I don't think that general fact is worth including on this page either. However, it may be worth including that (13/10)^2 is sharp of 27/16 by just [[676/675]].--[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 03:34, 19 October 2025 (UTC)
+: 567/320 looks complex because it has large numbers, but really it's just 7/5 above 81/64. Granted, it's more complex than 20/7 (which is 10/7 above 2/1), but it's certainly not an extremely complex septimal interval either.
+: The intuition behind this property is that common interseptimal intervals are rather well centered between the two septimal intervals that serve as boundaries for the region, and likewise that common septimal intervals are rather well centered between their nearest interseptimal region and their nearest diatonic category. For example, the M3-P4 spectrum includes 81/64 (Pythagorean), then 9/7 (septimal), then 13/10 (interseptimal), then 21/16 (septimal), then 4/3 (Pythagorean). Thus an interseptimal interval has roughly 1/4 of the distance between two diatonic categories of an error to its closest septimal interval. When brought to the fourth power, the error approaches a semitone, and therefore it ends up approximating another septimal interval.
+: 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)