Creating new EDO pages
Hi FloraC, I wanted to let you know that I was just filling in some gaps in EDO pages temporarily, I plan to expand the pages at some point in the future. MisterShafXen (talk) 15:10, 8 January 2025 (UTC)
En dashes vs. em dashes
Hi, can you point out on which article I replaced en dashes with em dashes where they were interchangeable? Was it on 3L 4s? Because if that's the page you were referring to, on that page I replaced some hyphen-minuses with minus signs, not en dashes with em dashes. Note that minus signs and em/en dashes can be hard to distinguish from each other in the source editor. [math]\displaystyle{ \overrightarrow{\,\,\scriptsize\text{ArrowHead294}\,\,\,} }[/math] (speak to me) 15:59, 25 February 2025 (UTC)
What exactly are the reasons for keeping "Pythagorean" but not allowing names for higher-limit trivial temps?
I personally find defining trivial temps useful as the families defined by them determine what temperaments "flatten" larger subgroups down to the trivial temperament's subgroup.
- First of all, the name of the trivial temps in all limits is JI.
- Pythagorean tuning is kept becuz it has history and becuz Pythagoras was known to haved worked with the 3-limit. Also, Pythagorean tuning isn't called pythagorean temperament. The latter should generally be avoided cuz it can be confused with compton, the temp that tempers out the Pythagorean comma. Another example is Alpharabian tuning, which was lately added cuz al-Farabi was known to have worked with undecimal intervals.
- Now, given Alpharabian tuning has been added, it would seem okay to add something like Classical tuning. After all, Classical music tempers 5-limit intervals, which give rise to the name classical intervals, so it's related. However, "Classical tuning" has a big problem: it can be used to contrast meantone, the temperament that tempers classical intervals and that is actually used by Classical music. Iow which of JI and meantone is more Classical? It's meantone. Therefore, Classical tuning isn't a proper alias for JI.
On declaring Zeta peaks
Hello Flora, regarding some of the changes I submitted on some EDO pages like the 217EDO or 311EDO page, and more generally, why not declare these big zeta EDOs to be such in the ET infobox, or in the case of 311EDO, why not expand the visible limit to 41? I feel like that'd be important information about these EDOs that would otherwise be left out to the average reader. --EufalesioEufalesio (talk) 10:56, 11 September 2025 (UTC)
- The community decided that zeta should not be displayed in the infobox some time ago, with some saying they don't care it being displayed in large edos. However, until a line is drawn somewhere, in which case it can be automated, I'm just gonna keep this feature out for consistency's sake.
- For the harmonics table, the default number of columns is usually good and extra columns should be added in a separate, collapsed table. Overwidth tables break the page layout on some devices.
On temperament degeneracy
Hello Flora, regarding the changes in concept I submitted to Bixby and Archon, why aren't they not degenerate?
For Bixby, if you temper out 4/3, it results in ~3/2 being mapped to an octave. If you temper 3/2 instead, 4/3 is 1 octave. If you temper 3/1 instead, then all 3's become unisons. In all of these cases, you are effectively destroying prime 3 and forcing it to be equal to some number of octaves, however as many as you like, so it matters little what you temper. Same with Archon destroying prime 5.
It would be the same thing as tempering out 17/16, tempering an octave-reduced harmonic to the nearest octave would be effectively removing it from the system assuming pure octaves. It should follow that the simplest subgroup of this temperament, 2.17, it is equivalent to the 2-limit, and can only be represented by 1edo; you get no real sonic benefit from just using octaves.
Running tuning optimizations on those is possible, but the data you get is wild because the optimizer is trying to make up for the fact that a prime would be mapped to pure octaves.
Antitonic wouldn't be a degenerate temperament because while 9/8 is a wild thing to be called a comma, it can be represented at least in 2edo and 4edo, and you get the tiniest semblance of 3/2... truly a troll temperament.
I would call degenerate any temperament that can only represented by patent val by 1edN, or worse by 0edo (which of course is trivial). --Eufalesio (talk) 12:30, 31 December 2025 (UTC)
- The octave can be tempered.
- Bixby isn't the same as the 2.5 subgroup. 2.3.5 is reduced to 2.5 only if you temper out 3/1. In bixby ~2/1 can be bent towards ~3/2 just as ~3/2 can be bent towards ~2/1. For the same reason archon isn't the same as the 2.3 subgroup.
- Tempering out 17/16 easily makes sense by detuning each octave 25 cents wide to get a 17/1 that's only about 5 cents narrow.
- Even tho these temps are still wild, they are not as wild as you think, and certainly not degenerate cuz degeneracy means actually removing a dimension, like tempering out a literal harmonic and not just selectively an octave-reduced harmonic.
- Degeneracy is normally understood to be limiting behavior; the case of degeneracy being a removal of a dimension is specific to linear algebra, according to the broader mathematical definition, these temperaments can still be considered degenerate cases, hear me out:
- Assuming 1edo is a trivial temperament of the 2-limit:
- Let's think about the simplest case scenario: Going from the 3-limit, tempering out 4/3 to get a rank-1 temperament. This "rank-1 Bixby" can only be represented by patent val by 1edo, where each step is an (tempered) octave, and fifths are rounded to octaves, same with 5/4s and all other primes. Here, the structure has collapsed, since only the 2-limit is left and only the trivial temperament (1edo) can represent it.
- Even in the 5-limit extension (Bixby proper), no other type of temperament collapses its structure so much that it is only representable by patent val by 1edo, not even Antitonic. Since the 3-structure has collapsed to the 2-limit and only the 2.5 subgroup makes sense, I believe that the only reasonable way to look at this, is to see it as a degenerate case.
- In Antitonic, you can temper the period to get a bit closer to "fifths" and "octaves", but the structure has not collapsed. In fact, Antitonic is the first proper 3-limit temperament, it even demonstrates 3-2 telicity!
- I believe this is a useful case of degeneracy that should be used to categorize these limiting temperaments. I wouldn't deem them exotemperaments, no that's far too generous... just degenerate temperaments.
- Degeneracy from wiktionary:
- > (mathematics) A limiting case of a class of objects which appears to be qualitatively different from (and usually simpler than) the rest of the class.
- I believe this is the broader mathematical definition you're talking about.
- As I said, the only way to reduce a 3-limit structure to 2-limit to make it appear qualitatively different is to temper out 2/1 or 3/1. Tempering out 4/3 isn't the same as tempering out 3/1 since 3/2 can be "rounded" to 2/1 to result in 1edo just as 2/1 can be "rounded" to 3/2 to result in 1edf. While the former is the same as the 2-limit, the latter is not, nor is anything in between. This is the behavior of an ordinary temperament and not a degenerate one.
- Furthermore, bixby is the rank-2 temp that uses the 1et for the 3-limit part but gives an independent generator for prime 5. The way it uses 1et for the 3-limit isn't different from how blackwood uses 5et for the 3-limit, and is perfectly functional. In fact bixby is supported by the following GPVs: ⟨1 2 2] (1), ⟨1 2 3] (1c), ⟨2 4 5] (2b), and ⟨3 6 8] (3bbcc). Note that 1c is equivalent to b2, the patent val of 2edt. 2b is the patent val of a slightly stretched 4edt, and 3bbcc is the patent val of a carefully stretched 3edo. So it's not like it's only supported by a single equal temperament. Even if it was, it would only qualify for degeneracy in a very peculiar sense.
- Therefore, it follows that calling bixby degenerate is not only conceptually unfounded, but takes away the verbal distinction of truly degenerate temps. Now I don't mind giving them a new category inside or alongside exotemps for their extreme inaccuracy (that would possibly include any temps less accurate than father) but I urge you to be careful with your choice of terms.
- Would you care continuing this discussion on Discord (among other things I'd also like to talk about with you)? I think this is going to get too long to put in the discussion page.
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