by Ted Argo

The study of consonance of musical sounds has primarily involved the relationship of one note to another. These interval relationships are governed by two different historically developed tuning schemes. The first, the Pythagorean intervals, are based off of whole number ratios between notes in a scale. For example, a fifth is a ratio of 3 to 2 to the fundamental or base tone. These values are shown in table 1 below.  The second method, also in table 1 below, is based on the twelfth root of two and is called equal temperament. For example, the third scale degree is the three-twelfth root of 2.

These categories are organized from the smoothest sounds to the most dissonant, as shown in Table 2.
The first degree consists of only the unison interval as these two waves align completely. The second degree consists of the octave, where the two waves are aligned such that one is exactly twice the frequency of the other. Then as the degrees of sweetness then increase the lowest common multiples of their ratios are organized according to a calculated naming scheme. The scheme that was used by Euler is generalized to a scheme of the multiples of the prime factors of the ratios as shown:
For a ratio of 1:k₁^p1:k₂^p2:…:k_n^pn
Degree=1+S(k_m*p_m-k_m)
For example, the fifth will be:(1*3-1)+(1*2-1) + 1= 4.
The fifth is found at Euler's degree of sweetness four.
The problem with his method lies in higher complexity chords. Given any higher degree chord, for example a seventh chord, the lowest common multiple fits into Euler's table properly, but the scheme for deriving that place in the table breaks down. Major seven chords consist of a root, major third, fifth, and major seventh. The math according to Euler is as follows:
LCM(1:8:10:12:15)=120
Factorization:1, 2³, 2*5, 3*2², 3*5
Simplifies to: 1, 2⁶, 3², 5
Degree =(1*1-1)+(6*2-6)+(2*3-2)+(2*5-2)+1
=0+6+4+8+1=19
This chord is supposed to fit into degree 10, but seems to fit into degree 19 when calculated.
This is only one of the many ways people of all disciplines have looked at the science of musical structure. Despite all of the mathematical investigation, a method has not been found to

When talking about musical structures, the ratios between the various scale degrees and the tonic is important. For a major fifth, the 3:2:1 ratio defines the relationship. For something like a major chord (the root, major third, and perfect fifth) the relationship is a 6:5:4:1 ratio. (3:2 is converted to 6:4 so that the tones are of the same denominator)
Leonhard Euler categorized chords of two or more sounds into what he deemed "degrees of sweetness". His categorization is based on the lowest common multiples of the scale degrees of the notes within the chord and putting them into specific categories.