User:A/Markov constant

The golden ratio is well-known to xen obsessives for its high degree of dissonance when use as a frequency ratio (acoustic phi). The golden ratio also has a mathematical property that its continued fraction convergents approach it at an unusually slow rate, so it's "hard to approximate with JI ratios." It's presumptuous to say that the dissonance may be linked to this property, but it's pretty interesting nonetheless.

We can investigate further by bringing in the broader idea of Markov constants from Diophantine approximation theory, which provide a method to measure how difficult it is to approximate a certain irrational number with rationals. Every irrational number has a Markov constant [math]\displaystyle{ M(x) }[/math] which is either infinite or a positive real number. [math]\displaystyle{ M(x) }[/math] is finite iff [math]\displaystyle{ x }[/math]'s simple continued fraction is bounded.

If [math]\displaystyle{ y = \frac{ax + b}{cx + d} }[/math] with integers [math]\displaystyle{ ad - bc = \pm 1 }[/math], then [math]\displaystyle{ M(x) = M(y) }[/math] and [math]\displaystyle{ x }[/math] and [math]\displaystyle{ y }[/math] are equivalent in Diophantine approximation. Transformations of the form [math]\displaystyle{ x \mapsto \frac{ax + b}{cx + d} }[/math] with integers [math]\displaystyle{ ad - bc = \pm 1 }[/math] form a group, which if my Wikipedia understanding of the modular group is accurate, is generated by the four transformations [math]\displaystyle{ x \mapsto -x }[/math], [math]\displaystyle{ x \mapsto 1 \pm \frac{1}{x} }[/math], and [math]\displaystyle{ x \mapsto \frac{1}{x \pm 1} }[/math].

Smaller [math]\displaystyle{ M(x) }[/math] have slower convergence in their continued fractions. We can treat [math]\displaystyle{ M(x) }[/math] as a somewhat sideways measure of "consonance," although there are no guarantees that it bears any relationship with actual consonance. The irrational numbers with the smallest Markov constants, and are therefore "most dissonant" by this measure, are:

Phi-equivalent intervals: [math]\displaystyle{ \phi }[/math] and all its equivalents share the minimum possible Markov constant at [math]\displaystyle{ M(\phi) = \sqrt{5} }[/math].

Tritone-equivalent intervals: [math]\displaystyle{ M(\sqrt{2}) = 2 \sqrt{2} }[/math] and its equivalents are in second place. It's pretty astonishing that, in a search for dissonance, the classically dissonant tritone naturally appears.

Minor sixth-equivalent intervals: In third place is [math]\displaystyle{ M\left(\frac{1 + \sqrt{221}}{10}\right) = \frac{\sqrt{221}}{5} }[/math] and all its equivalents. The constant [math]\displaystyle{ \mu = \frac{1 + \sqrt{221}}{10} }[/math] is about 799 cents or almost exactly a 12edo minor sixth, which is not classically considered a very dissonant interval.