Talk:Fifthspan

About ringy confusion

The word ringy looks confusing, at least from my perspective: it's not clear why it should mean multi-ring instead of single-ring. Wouldn't it be better to stick to the single-ring vs. multi-ring distinction? --Xenwolf (talk) 16:52, 20 November 2020 (UTC)

Ringy applies to multi-ring edos not single-ring edos for the same reason you usually don't refer to a room in a single-story house as being on the ground floor. If you're talking about floors, you're talking about a multi-story house. If you're talking about rings, you're talking about a multi-ring edo. Because in a single-ring edo, the subject never comes up.

That said, "ringy" is not as obvious as "multi-ring", and I don't mind removing most uses of it in the fifthspan article. But I do want to mention the word once. Because even though we use language to communicate precise scientific/mathematical ideas, it's also true that language itself is an artform. IMO it actually shares a lot with music, as it has rhythm and melody, and timbre is supplied by vowel sounds. Ringy is concise, memorable, catchy, and fun to say. It's like a hooky pop chorus. It would be a shame not to include it in the article. unsigned contribution by: TallKite, 20:36, 21 November 2020 (UTC)

Thanks a lot. I expected that there was a good explanation. The problem I have reported seems to affect mainly people who are not native English speakers. I think the changes you made removed the confusion. --Xenwolf (talk) 20:49, 21 November 2020 (UTC)

Fifthward and fourthward distances for multi-ring EDOs?

It occurred to me that it should be possible to define fifthward and fourthward distances for multi-ring EDOs if fractional distances are allowed. For instance, 34EDO has 2 rings of fifths, so with only integer fifthwards and fourthwards distances, the fifthwards/fourthwards antipode is undefined. But if we divide the perfect fifth (20\34) in 4 (5\34), we get the odd number of 1\34 increments we need to move between rings of fifths. Then we can define a fifthward distance for any interval in 34EDO, for instance its best approximation to 5/4, which is 11\34: Moving upwards by 9 quarter-fifths (9/4 fifths) gives us 45\34, which octave-reduces to 11\34. Since Tetracot temperament divides the (tempered) perfect fifth into 4 equal parts, this provides a reason in favor of using the Tetracot generalized/isomorphic keyboard mapping for facilitating use of the excellent 5-limit harmony of 34EDO.

(Also see Talk:Antipodes for another example of potentially useful fractional distances.)

Lucius Chiaraviglio (talk) 21:37, 20 June 2024 (UTC)

This is a fascinating idea, but there's a problem. In single-ring edos, we have negative fifthspans, which are in effect fourthspans. But in multi-ring edos, using your method, the fourthspans are completely unrelated to fifthspans. Look at 24edo:

fifthspans: 0 7 14 21 4 11 18 1 8 15 22 5 12...

fourthspans: 0 5 10 15 20 1 6 11 16 21 2 7 12...

Thus 1\24 has a fifthspan of 3.5 (or -8.5), but a fourthspan of 2.5 (or -9.5). Using fifthspans, 5\24 is quite remote, but using fourthspans it's very nearby. I suppose we could make a rule that fourthspans are not allowed, but that seems capricious to me. --TallKite (talk) 21:59, 29 June 2024 (UTC)

Based upon the Ploidacot explanation, it shouldn't be a surprise that a fourthspan wouldn't work the same way as a negative fifthspan, since to get a fourthspan from a negative fifthspan, you need to add an equave, which might not be dividible by the same number as the fifthspan. In 24EDO you can divide both by 2, but in 34EDO, you can divide the fifth (20\34) by 4 (as part of 34EDO being tetracot), but you can't divide the fourth (14\34) by 4. So you would just have to accept that fifthspan and fourthspan won't have a simple relationship to each other.

Lucius Chiaraviglio (talk) 12:48, 8 July 2024 (UTC)

For instance, Mothra would be tricot, with the fifthspan to 7/4 being -1/3, and it would also be beta triquat (taking that for the moment as the fourths counterpart to tricot), with a double-octave-compounded-fourthspan to 7/4 being +1/3. Either way works the same (even on a fraction-named meantone such as quarter-comma meantone or even extended-limit Pythagorean tuning) as long as you don't try to apply it to an EDO having a fifth not divisible by 3 and demand that it produce an integer number of steps.

Lucius Chiaraviglio (talk) 06:12, 1 August 2024 (UTC)

Actually, thinking about combining this with the Ploidacot/Ploidaquat concept reveals that it works even better than I thought. To take the 24EDO example, 24EDO is (haploid) dicot (with no boost prefix), so dividing the fifth (14\24) in half doesn't need any additional ploidacot/ploidaquat actions, and +3.5 fifths = +3.5 * 14 = 49\24 which octave-reduces to 1\24, as advertised; and -8.5 fifths = -8.5 * 14 = -119\24 which octave-reduces to 1\24, as advertised. But since 24EDO is dicot, it is thereby also (haploid) alpha-diquat, which means that you have to add a ploid (24) to the fourth (10\24) before performing the next steps, so the alpha-boosted fourth is 34 increments. Then we do the multiplication by the fractional number, and -3.5 alpha-boosted fourths = -3.5 * 34 = -119\17 which octave-reduces to 1\24, just like +3.5 fifths; and +8.5 alpha-boosted fourths = 289\24 which octave-reduces to 1\24, just like the -8.5 fifths.

This works moving along the Dicot temperament, which is (haploid) dicot and (haploid) alpha-diquat and also includes the patent val of 17EDO, where the same thing works for +3.5 fifths (+3.5 * 10\17 = 35\17 octave which reduces to 1\17) and -3.5 alpha-boosted fourths (-3.5 * 24\17 = -84\17 which octave-reduces to 1\17). Of course, since 17EDO is odd, the progression in the reverse direction will come out to a whole number: -5.0 * 10\17 = -50\17 which octave-reduces to 1\17; and +5.0 * 24\17 (remember the alpha-boost to the fourth) = 120\17 which octave-reduces to 1\17.

The Mothra example I gave above works the same way, but Mothra is tricot and beta-triquat, so you have to double-boost the fourths before performing further steps. And the Tetracot example that I fumbled above before I had this whole thing worked out functions the same way, but Tetracot is tetracot and gamma-tetraquat, so you have to triple-boost the fourths before performing further steps. This even works for a polyploid tempermant like Blackwood, which would be tricot and beta-triquat if not for the rule that explicitly declares it to be acot for having the fifth constituted of a whole number of ploids.

Lucius Chiaraviglio (talk) 06:08, 3 August 2024 (UTC)

Edit for above: Forgot to multiply by ploidy for Blackwood, so if you bypass the rule that explicitly declares it to be acot, it would be tricot and iota-triquat. Lucius Chiaraviglio (talk) 18:46, 1 September 2024 (UTC)