I have explored the mathematics of the theoretical models of various equal temperaments and unequal temperaments.

However, the biggest drawback of all of these theoretical models is that they do not effectively account for inharmonicity.

Therefore, I consider inharmonicity to be the final piece of the puzzle.

I have been asking myself the following question:

How useful are the mathematical analyses of the theoretical models of various temperaments if they do not effectively account for inharmonicity?

I believe that inharmonicity should be integrated into the theoretical models mentioned above to make them more useful than they currently are.

For example, pure 12th equal temperament is based on the pure twelfth (frequency ratio: 3/1). Inharmonicity will cause the 3rd partial of the fundamental frequency to be sharper than the pure twelfth. Therefore, the pure twelfth will need to be sharpened/stretched/widened so that it matches the 3rd partial of the fundamental frequency. This means that

*pure 12th equal temperament* should actually be referred to as

*equal temperament based on the 3rd partial of the fundamental frequency* to account for inharmonicity. Is this correct?

It is likely to be difficult to effectively incorporate inharmonicity into the numerous theoretical models of unequal temperament since the inequality of their intervals is likely to complicate matters further than they already have been by inharmonicity.

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Roshan Kakiya

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