Concerning just intonation scales, this paper boils down to saying that, if a given ratio appears in a scale, then ratios with prime factors removed tend to appear in the same scale.
Since intervals whose ratios' numerators and denominators high LCMs tend to sound dissonant, it is not surprising that these complex ratios are present only when the simpler ratios are already "taken". Seems like the same evolutionary processes that result in languages with simple speech sounds may be at work here.
As far as even-tempered scales go, this is silly.
From section 2.2, on reducing even-tempered scales (think, transcendental numbers) to just intonation scales (ratios):
> In an Euler-lattice [...], unison vectors can be found which represent very small ratios,
...of COURSE if you approximate the tones with very small ratios, they will likely form a convex lattice, since very small ratios are all near the center of the lattice!
> of COURSE if you approximate the tones with very small ratios, they will likely form a convex lattice, since very small ratios are all near the center of the lattice!
I don't think that's what that part is saying. Small ratios i.e. ratios close to 1 like 16/15, 24/25, 27/25 don't appear near the center. Simple ratios such as 3/2 etc are the ones that appear near the center. If I understand it correctly this section is simply describing a transformation that collapses several points into one, equivalent to collapsing several similar notes into one.
It should also be pointed out to non-musicians that the chance that you have ever in your life heard music that uses a just intonation scale, which this study is restricted to, is very small.
I dunno, if you ever played a brass instrument, then you're playing on an (approximately?) just-tempered scale -- each valve combination is a harmonic series starting from the 2nd harmonic. Whether it's just-tempered over the entire scale depends on the way the valves are tuned, I guess, and it's likely that the tuning reference is equal-tempered most of the time, so it's kind of a hodgepodge. It probably has a name.
I play brass (trumpet) and was always warned on assuming the tuning was accurate; even with the perfect player using the perfect compensation on the relevant valves the lower notes are definitely iffy (they tend towards flat and the compensation slides can't push them sharp).
That perfect player is improbable though. On the lower notes I can bend them by in the order of a quarter tone without touching the valves. If I pick up the instrument on a cold day without first warming it, not just me, up then it will be probably a quarter tone or so flat for the first few minutes. If you leave it on your chair and come back to find someone's turned the heater on while you were away (happened to me before), it can easily be a semitone sharp, IF it's cool enough to hold in the first place!
I love my trumpet but it's one of the last instruments I'd use to illustrate really precise tuning.
> This is used to lower the pitch of the 1-3 and 1-2-3 valve combinations.
In other words, the tuning slides described in that section of the article are solving a different problem (which is described in a bit more detail in the previous section). As a1k0n said, a given valve combination is a harmonic series. If you play a major third on a trumpet without moving your hands, the distance between the notes will be 386 cents, not 400. Of course, a good player will do whatever they need to do to sound in tune with whoever they're playing with. My understanding is that the lips are the main thing used to get harmonics of a particular valve combination in tune with each other.
Since intervals whose ratios' numerators and denominators high LCMs tend to sound dissonant, it is not surprising that these complex ratios are present only when the simpler ratios are already "taken". Seems like the same evolutionary processes that result in languages with simple speech sounds may be at work here.
As far as even-tempered scales go, this is silly.
From section 2.2, on reducing even-tempered scales (think, transcendental numbers) to just intonation scales (ratios):
> In an Euler-lattice [...], unison vectors can be found which represent very small ratios,
...of COURSE if you approximate the tones with very small ratios, they will likely form a convex lattice, since very small ratios are all near the center of the lattice!