Talk:159edo: Difference between revisions

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: Now that I think about it, this section doesn't make sense. It's not about having the intervals be convincing as consonances, but having the system act similarly to JI, and intervals that are complex in terms of their ratios are often used.--[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 21:04, 16 October 2025 (UTC)

:: Telicity matters more for navigational and modulatory purposes than anything, and indeed, having complex 3-limit intervals act similarly to JI while the circle of fifths still closes on the octave is very useful for that reason. Also, telicity is a concept that was devised before consistent circles, and it actually affects mappings of higher primes relative to lower primes in general, so it's not just stuff along the Pythagorean chain. For instance, because three instances of 11/8 octave-reduced add up to something that's really close to a stack of four Pythagorean Chromatic Semitones, that helps make the 2.11 chain more navigable. Unfortunately, with EDOs, sometimes you're forced to pick some kinds of telicity over others for practical reasons. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 21:56, 16 October 2025 (UTC)

: I think I get why it matters now. Someone who is accustomed to 12edo will find notes up to ~50 cents apart, like 7/4 and 12/7, to sound similar, while notes further apart, like 5/4 and 6/5, sound completely different. If someone is accustomed to 159edo, and can distinguish all of its intervals, then this threshold drops to half a step of 159edo, around 3.7 cents so inconsistency is ''always'' a problem. The circle of fifths is crucial to western music theory, so 3-2 telicity is important in that regard. The article mentions intervals like 49/32 and 35/32 being inconsistent, but fortunately there aren't many chords involving these with concordant structures, unlike 55/32 which appears in chords like 32:40:48:55, which ''is'' mapped consistently. The condition of telicity quickly becomes restrictive in larger EDOs, though some like 159edo manage to slip through. Its very lucky that prime 11 specifically is tuned well in 159edo, and I personally find it to sound cool and perfectly consonant, while prime 7 doesn't feel as cool (though still consonant). Also, 159edo is distinctly consistent to the 17-odd-limit, which is at the limit for an EDO this size, with only 149edo before it also being distinctly consistent to the 17-odd-limit. 159edo fails 19-odd-limit consistency because 19/17 is mapped to 25 steps while it is closer to 26, though even in the no-17 19-limit it equates intervals like 19/15 and 24/19, which doesn't occur in lower odd limits, showing that you can't expect an EDO of this size to do so well past the 17-limit. An EDO with 2-3-11 telicity and similar or better properties doesn't occur for a while, by which point you might as well use JI. --[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 05:06, 15 November 2025 (UTC)

:: Your assessment is almost on the nose. Inconsistency is always a problem, so telicity offers a way to prune off the inconsistent intervals. You are right about 3-2 telicity being crucial to Western music theory, and thus, the kinds of music I write, but 5-3 telicity is also important for this kind of music, although to a lesser extent, and of course, 11-3 telicity is needed for good quartertones. The only real quibble I have with your assessment is that it should be noted both that even a system like 159edo is vastly simplified in terms of interval arithmetic compared to JI, and that the just-noticeable difference between pitches, which is situated at about 3.5 cents, is another significant practical restriction against large EDOs as well as large prime limits. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 06:16, 15 November 2025 (UTC)

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 : Now that I think about it, this section doesn't make sense. It's not about having the intervals be convincing as consonances, but having the system act similarly to JI, and intervals that are complex in terms of their ratios are often used.--[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 21:04, 16 October 2025 (UTC)
+:: Telicity matters more for navigational and modulatory purposes than anything, and indeed, having complex 3-limit intervals act similarly to JI while the circle of fifths still closes on the octave is very useful for that reason.  Also, telicity is a concept that was devised before consistent circles, and it actually affects mappings of higher primes relative to lower primes in general, so it's not just stuff along the Pythagorean chain.  For instance, because three instances of 11/8 octave-reduced add up to something that's really close to a stack of four Pythagorean Chromatic Semitones, that helps make the 2.11 chain more navigable.  Unfortunately, with EDOs, sometimes you're forced to pick some kinds of telicity over others for practical reasons. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 21:56, 16 October 2025 (UTC)
+: I think I get why it matters now. Someone who is accustomed to 12edo will find notes up to ~50 cents apart, like 7/4 and 12/7, to sound similar, while notes further apart, like 5/4 and 6/5, sound completely different. If someone is accustomed to 159edo, and can distinguish all of its intervals, then this threshold drops to half a step of 159edo, around 3.7 cents so inconsistency is ''always'' a problem. The circle of fifths is crucial to western music theory, so 3-2 telicity is important in that regard. The article mentions intervals like 49/32 and 35/32 being inconsistent, but fortunately there aren't many chords involving these with concordant structures, unlike 55/32 which appears in chords like 32:40:48:55, which ''is'' mapped consistently. The condition of telicity quickly becomes restrictive in larger EDOs, though some like 159edo manage to slip through. Its very lucky that prime 11 specifically is tuned well in 159edo, and I personally find it to sound cool and perfectly consonant, while prime 7 doesn't feel as cool (though still consonant). Also, 159edo is distinctly consistent to the 17-odd-limit, which is at the limit for an EDO this size, with only 149edo before it also being distinctly consistent to the 17-odd-limit. 159edo fails 19-odd-limit consistency because 19/17 is mapped to 25 steps while it is closer to 26, though even in the no-17 19-limit it equates intervals like 19/15 and 24/19, which doesn't occur in lower odd limits, showing that you can't expect an EDO of this size to do so well past the 17-limit. An EDO with 2-3-11 telicity and similar or better properties doesn't occur for a while, by which point you might as well use JI. --[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 05:06, 15 November 2025 (UTC)
+:: Your assessment is almost on the nose.  Inconsistency is always a problem, so telicity offers a way to prune off the inconsistent intervals. You are right about 3-2 telicity being crucial to Western music theory, and thus, the kinds of music I write, but 5-3 telicity is also important for this kind of music, although to a lesser extent, and of course, 11-3 telicity is needed for good quartertones.  The only real quibble I have with your assessment is that it should be noted both that even a system like 159edo is vastly simplified in terms of interval arithmetic compared to JI, and that the just-noticeable difference between pitches, which is situated at about 3.5 cents, is another significant practical restriction against large EDOs as well as large prime limits. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 06:16, 15 November 2025 (UTC)