Concordance: Difference between revisions
Created page with "'''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[https://en.xen.wiki/w/Tenney_norm] * Wilson norm[https://en.xen.wiki/w/Wilson_norm] * Harmonic entropy[https://en.xen.wiki/w/Harmonic_entropy] == Harmonic Cross-Beating == {{UTBox..." |
|||
| (One intermediate revision by one other user not shown) | |||
| Line 7: | Line 7: | ||
* Harmonic entropy[https://en.xen.wiki/w/Harmonic_entropy] | * Harmonic entropy[https://en.xen.wiki/w/Harmonic_entropy] | ||
== Harmonic | == Harmonic cross-beating == | ||
{{UTBox|g_|Ground|7766ff}} | {{UTBox|g_|Ground|7766ff}} | ||
[[File:Harmonic cross-beating saw 20-6.png|thumb|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.]] | [[File:Harmonic cross-beating saw 20-6.png|thumb|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 uses a probability distribution for how likely an interval is to be perceived as a certain ratio, which | 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 === | === 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 | 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 === | === Psychoacoustics === | ||
My hypothesis is that 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. | 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 | 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. | ||
Latest revision as of 11:38, 27 October 2026
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:
Harmonic cross-beating
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.
