User talk:Zhenlige

Commas and EDOs

Hello! I was just checking your EDO impressions page, I see you're the first person to jot down impressions of EDOs in the hundreds since I've added my own EDO impressions to this site. I'm surprised that anyone bothered to jot down impressions of 118edo, 171edo, or 224edo before Flora could. I myself tend to compose with 159edo for a number of reasons, and, seeing from your main user page that you're a math nerd, I'm now wondering what insights you'd find in analyzing a comma that I found while checking over the structure of 159edo. This comma is 3489136640/3486784401 and has a monzo of [19 -20 1 0 3>. It proves to be the amount by which three instances of 729/704 fall short of 10/9, as well as the amount by which three instances of 880/729 exceed 225/128. The reason for me asking is that I'm looking for a descriptive name for this comma. --Aura (talk) 13:50, 10 June 2025 (UTC)

I am surprised and honored that my user pages have been viewed by such an expert. I am sorry, but generally speaking, I am not familiar with either very large EDOs or very complex intervals. I have written impressions for some (53n+12)-edos (118, 171, 224) because they are good schismatic EDOs and usually support its 7-limit extension pontiac, which is also a microtemperament, just like how (12n+7)-edos (19, 31, 43) are good meantone EDOs. Schismatic and meantone are kind of similar: they both have the same structure as 3-limit JI; they are efficient temperaments supported by many EDOs; in optimized tunings, they both flatten 3/2 a bit to make 5/4 better; they both have an EDO with near-just 3/2 (12 and 53), a better generally but overtempered EDO (19 and 118), and the sum of the previous two, with near-optimized tuning and supporting many other useful temperaments (31 and 171). I am particularly interested in schismatic because of my previous research on 53edo. By the way, all of my user pages are currently unstable and may contain unsure ideas or feelings. (If my English contains any mistakes, feel free to correct me.) --Zhenlige (talk) 16:42, 10 June 2025 (UTC)

It's totally fair that you're not familiar with very large EDOs or very complex intervals if for no other reason than there just being way too many of both to cover. That said, what's interesting about schismatic is that it has implications for more subgroups than just the 5-limit- rather, it also has significant implications for other subgroups involving the 5-limit. I'm curious, how familiar are you with certain 11-limit intervals like 11/8, 16/11, 11/10, 33/32, and, of course, 243/242? Although I'm not convinced that you're at all familiar with the last of those five, I ask because I nevertheless want to see how familiar you are with some of these intervals. --Aura (talk) 20:29, 10 June 2025 (UTC)

I usually think 11-limit intervals as “neutral intervals ± fractions of rastma (243/242)”, just like how NFJS represents them. (I am not satisfied with the RoT's of FJS and NFJS, though.) For example, 11/8 is sA4 − 1/2 rastma, 33/32 is sA1 − 1/2 rastma, and so on. For intervals involving both 11 and 5 or both 11 and 7, that makes less sense, because thinking 11/10 as “n2₅ − 1/2 rastma” does not clearly show its properties since “n2₅” is already a strange interval. For specific intervals there may be other viewpoints, e.g. 11/10 as “something between 10/9 and 12/11” or “near 1/3 of the interval 4/3”. Actually I rarely think such intervals independently, but rather as part of a larger structure. 13-limit intervals are very novel to me — actually I am even more familiar with 17 and 19 due to my early research on near-12edo intervals. --Zhenlige (talk) 02:35, 11 June 2025 (UTC)

Funny you should mention not being satisfied with the terms of the FJS and NFJS. The nomenclature that goes with Alpharabian tuning is actually more precise. For instance, 33/32 is an ultraprime while 11/8 is a paramajor fourth. In contrast, 729/704 is an infra-augmented prime, while 243/176 is an infra-augmented fourth- you might want to look up these more complicated 2.3.11 intervals if you're not already familiar with them. I should mention that in addition to 11/10, a type of submajor second, being roughly one third of 4/3, it actually differs from an ultra-augmented prime (think an apotome plus 33/32) by a schisma. This naturally means that in schismatic temperaments, these two intervals are enharmonic. Any thoughts on all this? --Aura (talk) 09:46, 11 June 2025 (UTC)