Talk:Chords of pajara
Duplicate chords
Duplicate chords occur if and only if both 0 and 0' are present in the chord, as you get a different voicing that's taken as a distinct chord by swapping everything from each period to the other. In these cases, we need to decide what to do. I suggest the rule that the interval in the other period should follow the interval in the first period of the same step count, and we should only keep the first chord found in this order. —FloraC (talk) 06:47, 24 January 2026 (UTC)
- In these duplicate chords, one can be turned into the other by replacing n with n' and n' with n for each note n or n' in the chord. Some chords actually have both 0 and 0' but don't have a duplicate, such as 0–0'–2–2', where the formula to get the duplicate chord just returns back 0–0'–2–2'. I actually designed the code so that the 0 comes before 0', 1 before 1', etc. so we can just keep the first rotation that appears in the table.--Overthink (talk) 02:29, 26 January 2026 (UTC)
Which chords exist
I think it doesn't have a page because jubilismic temperament is relatively inaccurate, but jubilismic chords actually exist, and I have an example: the 50/49 tempering of 1–6/5–7/5–12/7. This isn't an essentially just chord in the 7- or 9-odd-limit, as by this transversal, 7/5–12/7 is a 60/49 interval, which has an odd limit of 49. If the tritone were 10/7 instead, then 6/5–10/7 would be a 25/21 interval, which is also in a high odd limit.
Also, I found a chord essentially tempered by 7-limit pajara, that being the tempering of 1–10/9–5/4–7/5–14/9–7/4. The 1–5/4–14/9 part means that this chord must have 225/224 be tempered out. The interval between 7/4 and 10/9 (63/40 in JI) isn't a consonance even after marvel tempering, and since pajara maps it to 8/5 (and not any other consonance), the only option which pajara supports is to equate it to 8/5 by adding 64/63 to the essential tempering commas of that chord.
As for the others, I'm not quite sure. It would be better to complete the tables first, and remove the ones that don't exist.
Also, there's actually a tetrad in the 11-limit which is a plurichord, That being 7:9:10:11~1/(14:11:10:9).
--Overthink (talk) 02:17, 26 January 2026 (UTC)
- You're right. It seems both jubilismic and semaphoresmic chords exist in the form of tetrads. —FloraC (talk) 11:55, 26 January 2026 (UTC)
- The chord 1–10/9–5/4–7/5–14/9–7/4, that is hexad #1, can be tempered from apollo. From Chords of magic page it seems 7-limit and 11-limit magic chords are not distinguished, and I think we can follow that, so that pajara chords are those that require three independent commas to be tempered out, which means they don't exist. Also, altho minerva chords are given in Chords of orwell page, I noticed plurichords and det chords aren't distinguished there and that many of the minerva chords aren't really minerva det chords. I'll need to review that one to check whether such chords exist. It's difficult to work out from here cuz pajara conflates so many LCJI intervals. —FloraC (talk) 14:49, 30 January 2026 (UTC)
- Nvm all magic chords are 11-limit.
- And I'm starting to feel the chord types on this page are in an awkward state. To recap, minerva chords exist, except that all such chords in pajara can be tempered to apollo, and perhaps most apollo chords in pajara can be tempered to minerva too. Which type of chords we present largely depends on which individual intervals we choose to present, and as you can see the current setting leads to the result of leaving only one minerva chord, in the hexads section.
- I wonder if it makes more sense to set the priorities by types of chords than by individual intervals, so that by setting apollo > minerva we don't need to worry about the latter type here.
- I was just trying to make it fit; Apollo is actually higher damage than minerva, even though 100/99 is smaller than 99/98, but it wouldn't be obvious to the reader. But I guess having the less accurate ones come first makes sense. As for pentacircle vs marvel, it's right on the edge. Calculating TOP damage from (size in cents)/log_2(num * denom), marvel has a TOP damage of 0.49366 ¢, while pentacitcle has a TOP damage of 0.49412 ¢. The ordering might be different by metrics like minimum WE/CWE error. --Overthink (talk) 22:16, 18 February 2026 (UTC)
- There are chords essentially tempered by rank-2 miracle in the 11-limit (see miracle chords). I've been wondering for a while, are there any chords essentially tempered by a rank-1 temperament? Even further, are there chords which could by considered dyadically consonant by fudging, but tempering out all the corresponding commas is impossible without mapping everything to the unison?--Overthink (talk) 01:38, 1 February 2026 (UTC)
Typing chords
This sentence seems incorrect, and it should probably be changed.
If the chord is essentially tempered, it is analyzed in terms of the transversal that requires the minimum amount of commas to be tempered out; if there is a tie between multiple transversals, it is analyzed in terms of the transversal which employs as many as possible of 9/5, 8/7, 9/7, and 7/5 above the root; if there's still a tie, 7/5, 8/7, 9/5, and 9/7 are prioritized in that order.
By this rule, 1–9/7–8/5 would be just marvel, rather than marvel/valinorsmic as it's listed. By what rule did you follow, and how would this sentence need to be changed? --Overthink (talk) 17:03, 30 January 2026 (UTC)
- Technically the tempering required for the chord doesn't depend on which transversal is chosen to represent it, but I guess you have a good point. I shouldn't over-analyse them. One thing that does need to be changed is the priority of the intervals, where I usually prefer 9/7 and 9/5 to 14/11 and 20/11 first. —FloraC (talk) 11:13, 31 January 2026 (UTC)