Concordance - Revision history

Inthar: /* Harmonic Cross-Beating */

2026-10-27T11:38:39Z

Harmonic Cross-Beating

← Older revision		Revision as of 11:38, 27 October 2026
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	* Harmonic entropy[https://en.xen.wiki/w/Harmonic_entropy]		* Harmonic entropy[https://en.xen.wiki/w/Harmonic_entropy]

	== Harmonic ~~Cross~~-~~Beating~~ ==		== 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.]]

Ground at 03:05, 27 October 2026

2026-10-27T03:05:05Z

← Older revision		Revision as of 03:05, 27 October 2026
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	{{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 ~~is a choice I contend with~~. I created '''harmonic cross-beating''' to address these issues. This model doesn't use ratios at all, and the probability distribution is ~~for whether~~ a difference between harmonics is ~~heard~~ as ~~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 ===		=== 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 "beating ~~confidence~~ curve~~" that approximates the strength of audible beating~~. Finally, sum everything.		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 ~~really~~ 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 ~~audible~~ beating frequency.		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.

Ground: 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..."

2026-10-27T02:35:32Z

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..."

New page

'''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|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.]]
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 is a choice I contend with. I created '''harmonic cross-beating''' to address these issues. This model doesn't use ratios at all, and the probability distribution is for whether a difference between harmonics is heard as beating.

=== 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 "beating confidence curve" that approximates the strength of audible beating. Finally, sum everything.

=== 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.

Having the axes in frequency space rather than logarithmic space really 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 audible beating frequency.