User talk:Godtone: Difference between revisions

Line 110:

== "Augmented–chromatic equivalence continuum" ==

I'm gonna move that article to a subpage for now. First, the title is inappropriate cuz when I see augmented and chromatic I think about 128/125 and 2187/2048, which is a comma basis of 21et. If someone create an equivalence continuum about 21et that will be the title. Second, I don't get at all what's bad about the father--3 continuum and what's not bad about your augmented--dicot continuum. Father is the simplest comma of 3et with order 1 in the prime 5 and is marked against the 3-limit comma 32/27, just like any other continua. Your choice of commas looks arbitrary and unjustifiable to me. Even if you have valid reasons to justify it, it's prolly too deep and niche to quickly grasp compared to the obvious way to organize them. Third, there's the change of parameter used in some continua and you can do that in the father--3 article too. Just use a different variable than ''n'', ''m'', and ''k'', and give your reasons! [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 11:03, 27 February 2025 (UTC)

: Sorry, my mistake with "chroma", I was thinking that it refers to 25/24 based on the [[syntonic–chromatic equivalence continuum]] and based on me intuitively thinking of a chroma as being strictly smaller than a semitone so that I often feel that 2187/2048 is the odd one out. But in my defence, it was used before in this way on the [[Father–3 equivalence continuum]] page:

:: "[...] such that temperaments satisfy ('''25/24''')k = '''16/15'''. This gives rise to the name '''chromatic'''–'''diatonic''' equivalence continuum, where both chromatic and diatonic refer to the classical versions of semitones."

: Anyways, to explain the choice of commas:

: I'm pretty certain* that 128/125 and 25/24 is the best way to organize this continuum because 1\3 is ''relatively speaking'' an extremely good approximation of 5/4, so that practically all the most notable and useful 5-limit temperaments supported by 3edo are defined principally in terms of how the genchain of ~5/4's finds prime 3 (which I'll explain in a moment), as if I'm not mistaken, integer ''n'' (for (128/125)n / (25/24)) always corresponds to using ~5/4 as a gen, so that generally speaking (tho I'm unsure if this is always true) non-integer ''n'' corresponds to using a gen that splits 5/4, 8/5, 2/1 or an octave-equivalent correspondingly. That is, one reason for 128/125 is that 3edo is notable not for its "5-limit" but for its 2.5 subgroup which is also partly why having 5^2 present in the other comma is acceptable IMO because it will cancel with 5^3 (though I discuss an alternative at the end that you might prefer, though I'd want to see it first cuz I worry it'll lead to many weird temperaments at simple points drowning out the interesting ones).

: Let me quote from the page to explain the IMO very straightforward musical significance/application of describing the continuum in this way:

:: "This formulation has a specific reason: 128/125 is significantly smaller than 25/24, so that it makes sense to equate some number of 128/125's with 25/24, but because {{nowrap|25/24 {{=}} ([[25/16]])/([[3/2]])}}, this has the consequence of clearly relating the ''n'' in {{nowrap|(128/125)n {{=}} 25/24}} with how many 5/4's are used to reach 3/2 (when octave-reduced):

:: If ''n'' = 0, then it takes no 128/125's to reach 25/24, implying 25/24's size is 0 (so that it's tempered out), meaning that 3/2 is reached via (5/4)2.

:: For integer ''n'' > 0, we always reach 25/24 via (25/16)/(128/125)''n'' because of (128/125)''n'' ~ 25/24 by definition, meaning that we reach 3/2 at 3''n'' + 2 generators of ~5/4, octave-reduced."

: That is, ''n'' is the number of dieses you need to flatten ~25/16 by to reach ~3/2, which seems musically very clear to me, as this explanation tells you how many gens of ~5/4 you need to move to reach ~3/2 (up to octave-equivalence), as each diesis is three ~5/4's, and thereby tells you the intuitive logic of pumping the comma of the associated temp for integer ''n'' as well as gives you crucial information about how many intervals are guaranteed to exist between ~6/5 and ~5/4 in a nontrivial tuning (which is why the integer part is important; ''n'' - 1 is the number of intervals between 6/5 and 5/4, which is equivalent to saying ''n'' is how many pieces we split 25/24 into).

: Having said that, it'd be good to also include a change of basis ''k'' = 3''n'' + 2 so that we don't have to manually and indirectly deduce the number of gens of ~5/4 needed for ~3/2 (because this can be annoying for non-integer ''n''), but I don't know how to calculate the associated comma pair, but I'm pretty sure it doesn't result in any of the coordinate schemes currently documented in [[Father–3 equivalence continuum]]. Like, are you really okay with magic being fractional in both ''n'' and ''m'' there? Also, IMO, 25/24 being at 0 is super natural; other than large integer values of ''n'', it's the only natural number value of ''n'' that corresponds to an exotemp, and is meaningfully trivial compared to integer ''n'' > 0. Also notice how as ''n'' grows, ~5/4 becomes sharper and approaches 1\3, which makes taking the limit of ''n'' to infinity to reach ~5/4 = 1\3 very intuitive as well. This also makes ''n'' = 1 (magic) notable as the simplest nontrivial tuning (one that admits a reasonable amount of structure and as a result isn't an exotemp), and if you accept the approximation ~5/4 = 1\3 (which even I accept given the right tuning of the fifth), then every integer ''n'' > 0 is non-exo in terms of the tuning of the 5-odd-limit (even if the edo and mapping complexity implied is absurd).

: <nowiki>*</nowiki> Re: "pretty certain", initially I was completely certain, hence the boldness of my edit cuz it seemed like a very unambiguous and significant improvement, but the consideration about 3''n'' + 2 maybe being a little more work than is reasonable to ask of the user makes me unsure given I haven't seen what the continuum looks like if we apply this transformation. The alternative transformation would have dicot at 2, magic at 5, mutt at 7, würschmidt at 8, etc. but already we see this might not be desirable because the info that mutt splits something (the octave and the diesis) into 3 is lost.

@@ Line 110: / Line 110: @@
 == "Augmented–chromatic equivalence continuum" ==
 I'm gonna move that article to a subpage for now. First, the title is inappropriate cuz when I see augmented and chromatic I think about 128/125 and 2187/2048, which is a comma basis of 21et. If someone create an equivalence continuum about 21et that will be the title. Second, I don't get at all what's bad about the father--3 continuum and what's not bad about your augmented--dicot continuum. Father is the simplest comma of 3et with order 1 in the prime 5 and is marked against the 3-limit comma 32/27, just like any other continua. Your choice of commas looks arbitrary and unjustifiable to me. Even if you have valid reasons to justify it, it's prolly too deep and niche to quickly grasp compared to the obvious way to organize them. Third, there's the change of parameter used in some continua and you can do that in the father--3 article too. Just use a different variable than ''n'', ''m'', and ''k'', and give your reasons! [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 11:03, 27 February 2025 (UTC)
+: Sorry, my mistake with "chroma", I was thinking that it refers to 25/24 based on the [[syntonic–chromatic equivalence continuum]] and based on me intuitively thinking of a chroma as being strictly smaller than a semitone so that I often feel that 2187/2048 is the odd one out. But in my defence, it was used before in this way on the [[Father–3 equivalence continuum]] page:
+:: "[...] such that temperaments satisfy ('''25/24''')<sup>k</sup> = '''16/15'''. This gives rise to the name '''chromatic'''–'''diatonic''' equivalence continuum, where both chromatic and diatonic refer to the classical versions of semitones."
+: Anyways, to explain the choice of commas:
+: I'm pretty certain* that 128/125 and 25/24 is the best way to organize this continuum because 1\3 is ''relatively speaking'' an extremely good approximation of 5/4, so that practically all the most notable and useful 5-limit temperaments supported by 3edo are defined principally in terms of how the genchain of ~5/4's finds prime 3 (which I'll explain in a moment), as if I'm not mistaken, integer ''n'' (for (128/125)<sup>n</sup> / (25/24)) always corresponds to using ~5/4 as a gen, so that generally speaking (tho I'm unsure if this is always true) non-integer ''n'' corresponds to using a gen that splits 5/4, 8/5, 2/1 or an octave-equivalent correspondingly. That is, one reason for 128/125 is that 3edo is notable not for its "5-limit" but for its 2.5 subgroup which is also partly why having 5^2 present in the other comma is acceptable IMO because it will cancel with 5^3 (though I discuss an alternative at the end that you might prefer, though I'd want to see it first cuz I worry it'll lead to many weird temperaments at simple points drowning out the interesting ones).
+: Let me quote from the page to explain the IMO very straightforward musical significance/application of describing the continuum in this way:
+:: "This formulation has a specific reason: 128/125 is significantly smaller than 25/24, so that it makes sense to equate some number of 128/125's with 25/24, but because {{nowrap|25/24 {{=}} ([[25/16]])/([[3/2]])}}, this has the consequence of clearly relating the ''n'' in {{nowrap|(128/125)<sup>n</sup> {{=}} 25/24}} with how many 5/4's are used to reach 3/2 (when octave-reduced):
+:: If ''n'' = 0, then it takes no 128/125's to reach 25/24, implying 25/24's size is 0 (so that it's tempered out), meaning that 3/2 is reached via (5/4)<sup>2</sup>.
+:: For integer ''n'' > 0, we always reach 25/24 via (25/16)/(128/125)<sup>''n''</sup> because of (128/125)<sup>''n''</sup> ~ 25/24 by definition, meaning that we reach 3/2 at 3''n'' + 2 generators of ~5/4, octave-reduced."
+: That is, ''n'' is the number of dieses you need to flatten ~25/16 by to reach ~3/2, which seems musically very clear to me, as this explanation tells you how many gens of ~5/4 you need to move to reach ~3/2 (up to octave-equivalence), as each diesis is three ~5/4's, and thereby tells you the intuitive logic of pumping the comma of the associated temp for integer ''n'' as well as gives you crucial information about how many intervals are guaranteed to exist between ~6/5 and ~5/4 in a nontrivial tuning (which is why the integer part is important; ''n'' - 1 is the number of intervals between 6/5 and 5/4, which is equivalent to saying ''n'' is how many pieces we split 25/24 into).
+: Having said that, it'd be good to also include a change of basis ''k'' = 3''n'' + 2 so that we don't have to manually and indirectly deduce the number of gens of ~5/4 needed for ~3/2 (because this can be annoying for non-integer ''n''), but I don't know how to calculate the associated comma pair, but I'm pretty sure it doesn't result in any of the coordinate schemes currently documented in [[Father–3 equivalence continuum]]. Like, are you really okay with magic being fractional in both ''n'' and ''m'' there? Also, IMO, 25/24 being at 0 is super natural; other than large integer values of ''n'', it's the only natural number value of ''n'' that corresponds to an exotemp, and is meaningfully trivial compared to integer ''n'' > 0. Also notice how as ''n'' grows, ~5/4 becomes sharper and approaches 1\3, which makes taking the limit of ''n'' to infinity to reach ~5/4 = 1\3 very intuitive as well. This also makes ''n'' = 1 (magic) notable as the simplest nontrivial tuning (one that admits a reasonable amount of structure and as a result isn't an exotemp), and if you accept the approximation ~5/4 = 1\3 (which even I accept given the right tuning of the fifth), then every integer ''n'' > 0 is non-exo in terms of the tuning of the 5-odd-limit (even if the edo and mapping complexity implied is absurd).
+: <nowiki>*</nowiki> Re: "pretty certain", initially I was completely certain, hence the boldness of my edit cuz it seemed like a very unambiguous and significant improvement, but the consideration about 3''n'' + 2 maybe being a little more work than is reasonable to ask of the user makes me unsure given I haven't seen what the continuum looks like if we apply this transformation. The alternative transformation would have dicot at 2, magic at 5, mutt at 7, würschmidt at 8, etc. but already we see this might not be desirable because the info that mutt splits something (the octave and the diesis) into 3 is lost.