Mathematicians often speak informally about the relative simplicity of rational numbers. For example, musical intervals that correspond to simple fractions have less tension than intervals that correspond to more complicated fractions.
Such informal statements can be made more precise using height functions. There are a variety of height functions designed for different applications, but the most common height function defines the height of a fraction p/q in lowest terms to be the sum of the numerator and denominator:
height(p/q) = |p| + |q|.
This post will look at how this applies to musical intervals, to approximations for π, and the number of days in a year.
Musical intervals
Here are musical intervals, ranked by the height of their just tuning frequency ratios.
- octave (2:1)
- fifth (3:2)
- forth (4:3)
- major sixth (5:3)
- major third (5:4)
- minor third (6:5)
- minor sixth (8:5)
- minor seventh (9:5)
- major second (10:9)
- major seventh (15:8)
- minor second (16:15)
- augmented fourth (45:32)
The least tension is an interval of an octave. The next six intervals are considered consonant. A minor seventh is considered mildly dissonant, and the rest are considered more dissonant. The most dissonant interval is the augmented fourth, also known as a tritone because it is the same interval as three whole steps.
Incidentally, a telephone busy signal consists of two pitches, 620 Hz and 480 Hz. This is a ratio of 24:31, which has a height of 54. This is consistent with the signal being moderately dissonant.
Approximations to π
The first four continued fraction approximations to π are 3, 22/7, 333/106, and 335/113.
Continued fraction convergents give the best rational approximation to an irrational for a given denominator. But for a height value that is not the height of a convergent, the best approximation might not be a convergent.
For example, the best approximation to π with height less than or equal to 333 + 106 is 333/106. But the best approximation with height less than or equal to 400 is 289/92, which is not a convergent of the continued fraction for π.
Days in a year
The number of days in a year is 365.2424177. Obviously that’s close to 365 1/4, and 1/4 is the best approximation to 0.2424177 for its height.
The Gregorian calendar has 97 leap days every 400 years, which approximates 0.2424177 with 97/400. This approximation has practical advantages for humans, but 8 leap days every 33 years would be a more accurate approximation with a much smaller height.
What you say about the musical intervals is true for just intonation. Pianos, however, are nowadays tuned using 12 equal temperament tuning in order to be able to transpose from one key to another. Consequently, that is the system that predominates in Western music. As a result, the only interval that is exact as you show is the octave. All others are close, but not quite the same as these. Fortunately, the human ear usually can’t tell the difference. I’m guessing you already know all this. ;)