Talk:7/4

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7/4 in 94edo

I don't know if you know this, Xenwolf, but 94edo is pretty good for 7/4 as well. --Aura (talk) 18:28, 25 October 2020 (UTC)

Yeah, 76\94 ( == 38\47, 9702.213 cents), is only 1.39 cents above 7/4. In 47edo it's within the relative tolerance limit (7%), in 94edo it's not. It's definitely not a bad approximation (only off by 10.9% of a 1\94). At first, I hand-calculated this table. Now I have a little python program (which is unfortunately incorruptible!) that has 2 parameters: the interval itself and the threshold of error magnitude (both, rel and abs). The upper EDO bound is currently fixed to 200, but could be a parameter, the separation of the rel and abs thresholds would possible as well. I know this is not really an answer to your non-question, but maybe helps to better understand why 94edo is not in the list: this decision has nothing to do with musical critera but only with with he difficulty to formalize harmonic quality (or my lack of imagination). --Xenwolf (talk) 20:40, 25 October 2020 (UTC)

BTW: Do you find this kind of table useful? --Xenwolf (talk) 20:43, 25 October 2020 (UTC)

Even if the calculations of this program are right, as I'm sure they are, no program is incorruptible- because somewhere in the production chain, it's sourced from something made by fallible humans. Nevertheless, as I'm sure the calculations are right, I'll accept the remainder of your explanation. As to whether or not these tables are useful, I'd say they are ultimately only useful for the articles dealing with the prime harmonics and their octave compliments. That said, there needs to be an additional column added to the charts. Specifically we need to add the number of times the tempered interval in question can be stacked without the absolute error between the tempered stack and its just counterpart exceeding 3.5 cents, or half a step- whichever is smaller. --Aura (talk) 21:25, 25 October 2020 (UTC)

I read half a step as 50% of 1\edo? Concerning the prime harmonics (their octave complements share the exact same table), is it actually that indisputable? I know the concept of consistency but I find it questionable already in cases like 7/5: this interval can probably be used independently of the fact that it can be derived from 7/4 and 5/4; another example is the great approximation of 11/7 in 23edo. Whatever, would you say that tables would useful to you if there was this column with the amount of repetitions within the limit you described? --Xenwolf (talk) 21:53, 25 October 2020 (UTC)

You are correct in your reading of half a step as 50% of 1\edo. While it is true that 7/5 can be used independently of the fact that it can be derived from 7/4 and 5/4, consistency is my main concern with these intervals. In my own work, it seems that wherever the p-limits from which intervals like 7/5 are derived are poorly represented and or subject to contortion in any given EDO, the consistency of the derived intervals is called the into question and or multiple possible representations in a given EDO occur. --Aura (talk) 22:14, 25 October 2020 (UTC)

My end goal with including a table containing the number of times the tempered p-limit interval in question can be stacked without the absolute error between the tempered stack and its just counterpart exceeding 3.5 cents (or half a step- whichever is smaller), has more to do with mapping out the usable portions of the harmonic lattice for any given EDO. --Aura (talk) 22:19, 25 October 2020 (UTC)

I have the impression that we still have little experience in generalizing the multiplication of intervals, so the combinability of fifths is certainly undisputed, but on the other hand it is strongly oriented towards functional harmony since the Baroque. I don't know any other interval that could perform such an axis function, although I can imagine that one could try it with the third. --Xenwolf (talk) 22:45, 25 October 2020 (UTC)

Remember what I said about the 11-limit being mathematically derivable as an excellent representation for quartertones in terms of ratio simplicity? One of the implications of this excellent representation- particualarly in light of the way it plays out- is that the paramajor fourth (that is, 11/8) can- and indeed it does- perform an axis function where quartertones are concerned. I've since checked the 11-limit's representation of quartertones against that of the other rational intervals called "quarter tones" on Wikipedia's list of pitch intervals and found the 11-limit's 33/32 to be better than any of them in terms of ratio simplicity. (multiple comments combined and edited by --Aura (talk) 23:02, 25 October 2020 (UTC))