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I thought inharmonicity was basically when the partials of a complex wave don't belong to integer multiples of a fundamental frequency.

I thought when this happens to get a "dissonant" sound, e.g. a tritone on a piano.

I recently found the term "inharmonicity coefficient" and the authors in the paper said inharmoncity is related to how "clear" the sound is.

What is the difference between dissonance and inharmoncity?

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    The inharmonicity coefficient is probably specific to a model used in that particular paper, which is attempting to quantify inharmonicity in some way to achieve some goal. Instead of being a "real" physical property of sounds. Mind linking the paper?
    – Edward
    Commented Jan 26, 2021 at 2:51

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Dissonance of the type you are talking about is created when two or more notes are played at the same time. Many types of dissonance are heard differently based on context and culture, and therefore is at least partially subjective.

Inharmonicity is a property of a single note. It is an aspect of the timbre of an instrument. Sounds with a small amount of inharmonicity, like a piano note, can be clearly heard as notes that have a richer or possibly shimmering character. Sounds with a great amount of inharmonicity, like the sound of a tam-tam, have much less of a sense of pitch and have a more “noisy” character. Inharmonicity is objectively measurable.

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  • How is inharmonicity objectively measurable? "A small about" should probably be "a small amount". Can't fix it, as it's less than 6 characters.
    – Pyromonk
    Commented Jan 26, 2021 at 11:07
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I thought inharmonicity was basically when the partials of a complex wave don't belong to integer multiples of a fundamental frequency.

Yes - a harmonic series is sequence of frequencies each of which is an integer multiple of a fundamental. The usual meaning of 'inharmonicity' is the degree to which partials of a sound deviate from that harmonic series, or a situation in which that deviation is occurring.

I recently found the term "inharmonicity coefficient" and the authors in the paper said inharmonicity is related to how "clear" the sound is.

That doesn't represent any standard definition of inharmonicity, but it may still be a fair statement in itself - because, for example, a more harmonic sound give have a clearer impression of pitch.

What is the difference between dissonance and inharmonicity?

We've defined inharmonicity above. 'Dissonance' is a somewhat general word that refers to some sound or note that audibly doesn't 'agree' with another sound. It might have a number of meanings dependent on context.

For example, two notes that give a sensation of roughness when sounded together may be called dissonant; a particular note that is judged to need resolution may also be called a 'dissonance' (hence why a the interval of a perfect fourth can be called 'dissonant' in some cases, even though two notes a perfect fourth apart do not sound 'rough'). Dissonance can also be defined in terms of the sensation of beating/roughness heard between two simultaneous sine waves, and groups thereof - see https://sethares.engr.wisc.edu/consemi.html.

Is dissonance a type of inharmonicity?

Not really - for a start, 'dissonance' is the more general word, while 'inharmonicity' refers to something more specific.

But could inharmonicity give rise to a type of dissonance? Actually yes, it could. If we have an instrument with a very inharmonic series of partials - perhaps something with multiple dimensions of resonance, like a drum or bell - we might find that two of the partials are less than a minor third apart, and therefore give rise to dissonance according to the definition on Sethares' page linked above.

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