How the brain perceives and judges the consonance (or otherwise) of an interval has been the theme of many expert studies. The issue is still the subject of controversy. But, as acoustics scientist Ernst Terhardt wrote in 1976 in a letter to layman Heinz Bohlen: "In science (especially in 'musical science') governs the right of free speech." It was Terhardt's reaction to Bohlen's opinion that gestalt enhancement by combination tones and harmonics is a crucial factor for consonance.

Referring to this view, Bohlen wrote in his paper "13 Tonstufen in der Duodezime" (1978) in footnote 5 that he doesn't want to introduce yet another method to calculate the consonance ranking of intervals, but if one would simply set, for each interval considered, the number of possible combination tones in the graph below in relation to those that actually appear, one would arrive at a measure of consonance that would neatly agree with experience.

Reproduction of graph by courtesy of S. Hirzel Verlag, Stuttgart

The graph shows the development of combination tones up to the eighth order for an interval consisting of a fixed base frequency f1 and a continuously increased frequency f2. Combination tone coincidences mark the intervals that are known consonances, forming "harmonic series" with them and thus enhancing their Gestalt impression.

As promised, in his paper Bohlen abstains from actually performing the evaluation that he hints at. But since no complex math is required, it is tempting to see what the outcome might have been. The result is shown in the following graph. For better comparison with harmonic entropy graphs, the degree of consonance (vertical axis) grows here with the negative numbers, unison at - 100 (meaning 100 % consonant) being the reference. The intervals evaluated appear on the horizontal axis in the order of their span (ratio).

Consonance degree of intervals (in %) vs. interval ratio

Despite the fact that the point of departure for this consonance assessment is entirely different from harmonic entropy considerations, the results seem surprisingly similar, as a comparison with the next graph demonstrates.

© Paul Erlich

This graph is shown in Paul Erlich's contribution on harmonic entropy to Joe Monzo's TonalsoftTM encyclopedia of microtonal music theory.

But giving the similarity a second thought, it becomes much less surprising than it seemed to be. The ranking is to a high degree a property of the special set of numbers that represent the interval ratios. This becomes obvious when contemplating one more attempt to explain consonance[1]. It stems from the Dutch ENT physician Theodoor Emile ter Kuile (1875 - 1941). He theorized that "the larger the relative chord period, the less capable the soul to recognize it as a unit, and the less it can be considered a consonant chord...". The relative chord period, as he defined it, is the time until the complex amplitude of the chord repeats its pattern, set in relation to the period of the chords base tone. This relative chord period grows with the numbers that the ratios of chord's tones consists of, and ter Kuile's equation for a chord's consonance V is given by

with α,β,γ etc. representing the components of the chord and p being the total number of components. Thus for an interval with the ratio α/β ter Kuile's formula is simply

Here the initial graph is repeated, this time containing ter Kuile's numbers for comparison.

Consonance degree of intervals (in %) vs. interval ratio
Criteria:
Blue: Gestalt enhancement by combination tones (Bohlen)
Green: Relative chord period length (ter Kuile)

Disparities between the two results appear to be more or less negligible, despite huge differences in criteria, methods and backing theories. Hence it seems that there are several usable approaches to the ranking of intervals, but that agreement with experience is no proof that the theory behind them is based on a correct understanding of the nature of consonance.