1. 5

For me the article explained in a very intuitive way how frequencies are derived from keys on the piano. This can be applied as is to the conversion of MIDI note numbers to frequencies, e.g. for a software synthesizer.

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  2. 6

    I think this article is pretty deceiving. You would never tune piano like this. At the bottom, it states:

    This tuning is called twelve-tone equal temperament. It’s “the most common tuning system since the 18th century”

    However, pianos are not tuned using twelve-tone equal temperament. They’re stretch tuned. Some electric pianos are also stretch tuned, and any piano sample plugin will also be stretched tuned. And indeed, the Wikipedia article Piano Key Frequencies that’s linked at the bottom says:

    This list of frequencies is for a theoretically ideal piano. On an actual piano the ratio between semitones is slightly larger, especially at the high and low ends, where string stiffness causes inharmonicity, i.e., the tendency for the harmonic makeup of each note to run sharp. To compensate for this, octaves are tuned slightly wide, stretched according to the inharmonic characteristics of each instrument. This deviation from equal temperament is called the Railsback curve.

    (I also take issue with the phrasing “theoretically ideal piano”. An “ideal” piano that’s used for actual music will be stretch tuned.)

    Further reading: http://piano.detwiler.us/

    edit: some comments on Hacker News also discuss this: https://news.ycombinator.com/item?id=19305258