Concordance

Concordance (opposed to discordance) is a measure of how well harmonics of different notes in a chord line up. It is often equated to consonance (opposed to dissonance), although consonance is cultural rather than physical.

Conventional models

In order of complexity:

• Tenney norm[1]
• Wilson norm[2]
• Harmonic entropy[3]

Harmonic cross-beating

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This section is written from the perspective of User:Ground and may feature proposed concepts.

Harmonic cross-beating diagram of two saw waves from 55 to 880 Hz. The beating confidence curve has a mean of 20 Hz and a standard deviation of 6 Hz.

Harmonic entropy is the accepted standard for quantifying concordance, but it does not explicitly take register and timbre into account. It also uses a probability distribution for how likely an interval is to be perceived as a certain integer ratio, which may not be the most perceptually significant metric. I created harmonic cross-beating to address these issues. This model doesn't use integer ratios at all, and the probability distribution is a "perceptual beating curve" that approximates the strength of audible beating in a difference between harmonics. I originally assumed a normal distribution with a mean of 20 Hz and a standard deviation of 6 Hz, which is about right as far as my experimentation goes.

Algorithm

Calculate the frequency difference between each harmonic of each waveform and each harmonic of the other waveforms to get a set of all possible beating frequencies. Multiply them by the relative strength of the current two harmonics being evaluated. Then multiply them by a perceptual beating curve. Finally, sum everything.

Psychoacoustics

My hypothesis is that the perceived dominant beating frequency in an interval with a given register and timbre is similar between listeners and can be mathematically modeled to high confidence, which can be used to approximate the vague consensus of concordance using a perceptual probability distribution.

Having the axes in frequency space rather than logarithmic space shows how the concordance wells have a constant width in terms of frequency, which is what I predicted, and new concordance wells gradually show up as the frequencies increase. The cross-beating values become unusually large as the waveforms approach the assumed perceptual beating frequency.