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Many of us are taught that the "complexity" of the ratio between two frequencies predicts its dissonant qualities. Is there a way to find a numerical value for dissonance? I have created a method, but I don't have the technology to test it.

My method is as follows: Take a = frequency 1, b = frequency 2

Ratio = a/b Dissonance = LCM(a, b) where LCM(x, y) is the least common multiple between two values x and y

This is rather simple, but I cannot find any mention of it online: I suppose it is either wrong or considered a given.

Take a few examples of different intervals' "dissonance":

Unison: 1/1 --> 1 Octave: 2/1 --> 2 Perfect Fifth: 3/2 --> 6 Major Third: 5/4 --> 20 Triple Octave: 8/1 --> 8

As a note, while doing research on the subject, I am attempting to remain in the realm of the non-extensive overtone series, and avoiding delving into synthetic ratios created through equal temperament and other tuning styles.

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  • You might want to look at William Sethares's work: sethares.engr.wisc.edu/ttss.html. You'll need to decide on a precise and measurable definition of "dissonance" rooted in whatever musical practice you're seeking to explain.
    – helveticat
    Commented Apr 18, 2022 at 10:19

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This is a rather difficult question and has no simple answer. One simple problem is that the concept of dissonance isn't easy to pin down. In many cases, it's context-dependent. Even just considering two notes in isolation, the interval of a fourth is troublesome; it has a simple ratio, 4/3, which happens to be the octave-reduced reciprocal of the fifth, 3/2, but in many situations, it's considered dissonant. In a chord of three or more notes, the fourth is dissonant if occurring against the bass, but consonant between two upper notes. The chords C-E-G and E-G-C are considered consonant but G-C-E is considered dissonant. When these are reduced to two-note textures, the G-C interval is dissonant with G in the bass but consonant otherwise.

https://www.frontiersin.org/articles/10.3389/fpsyg.2018.00381/full https://pubmed.ncbi.nlm.nih.gov/19048722/

A couple of somewhat psychologically-based references.

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