I was reading about spectral techniques to detect pathological phonation and came across the plan to establish relations between harmonic-harmonic and harmonic-noise ratios in for certain kinds of dysphonia. But what do these ratios have to do with voice pathologies?
As you know from your previous questions on the topic, the waveform of a breathy voiced vowel is more sinusoidal in shape compared to that of a modal voiced vowel, which is more triangular. The power spectrum of the two types of vowels differ in that more of the energy is distributed over low frequencies in breathy vowels, compared to modal vowels. As I recall, Rothenberg has characterized this as saying that the main difference is in the amplitude of the fundamental, where a breathy vowel has a very prominent fundamental and higher harmonics are much lower in amplitude, and flatter, but modal vowels have a more consistent gentle roll-off across frequencies. By comparing the ratio of two harmonics, one can quantify this relation.
An obvious technique for quantifying spectral tilt would be regression through amplitude of harmonic peaks. However, this does not work the way you'd expect, since a breathy spectrum is a high fundamental and everything else flatter, and regression essentially treats the fundamental as an outlier. Comparison of H1 and H2 has been the most useful metric since it involves the fundamental, and minimizes the effect of formant differences. There is evidence that planned linguistically-contrastive phonation differences can involve a lengthening of the vocal tract via lowering of the larynx, which leads to formant shifts, which gives you a bit of an unsolvable problem. However, for pathological breathiness, this is probably not relevant.
Harmonic to noise ratio calculation is based on the observation that with breathiness, there is a lot of non-harmonic noise in the signal (though this is not always true with linguistic breathiness), where the amplitude of the spectrum at a multiple of the fundamental doesn't stand out so distinctly.
I don't know whether research on the topic has concluded that there are any reliable acoustic differences between pathological breathiness and phonemic breathiness. (Likewise, creakiness).
[EDIT] Harmonic-to-harmonic measures are supposed to quantify the concept of "spectral tilt", which relates to the overall waveform shape which in turn relates to the glottal volume velocity function which relates to how the vocal folds open and close. The ideal (in the sense of an ideal frictionless surface) is that there would be some amplitude at the fundamental, and other amplitudes at multiples of the fundamental, and nothing in between. However, that ain't the reality of it, so besides the energy contributed by the vibration of the vocal folds, there is "noise", which is low-level random energy (which may have a particular frequency distribution). So then you need a way of discerning the "noisiness" of the signal. As applied to phonation types, as far as I know the added noise should be evenly distributed across frequencies, so you don't expect concentrations of high-frequency noise as you get with [s]; nor should the noise be just low-frequency. So there is a difference in expectation, compared to measures of spectral tilt.
I'm not sure what you mean by "harmonic-to-harmonic ratio" versus "harmonic-to-frequency ratio".