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Comment by amelius | Hacker News Reader

> There are twelve semitones in every octave and they make up just about all the music we hear in the West today.

Is it just me, or is this strict division into semitones making e.g. piano music sound like it's slightly off-key at times?

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Yep. There are many different ways to tune a piano, none of them perfect. A tuning that is just right for one key will leave real howlers in others. (Literally "howlers"; it's called a "wolf tone")

https://en.m.wikipedia.org/wiki/Interval_(music)

Actual pianos are usually tuned in a way that sounds pretty good most of the time, without ever sounding perfect.

“Well-tempered” (yes, like the clavier).

Pianos are tuned differently, they use stretch tuning and will be pretty out of tune with other instruments at both lowest and highest notes (https://en.wikipedia.org/wiki/Piano_tuning#/media/File:Rails...)

It's the other instruments that are out of tune.

HN discussion of "The Saddest Thing I know about the Integers"

https://news.ycombinator.com/item?id=8682782

Original article is offline, it seems :-(

EDIT: archive.org to the rescue!

https://web.archive.org/web/20240123154411/https://blogs.sci...

Is there a video that demonstrates the difference?

It's not short, but I was introduced to the concept in a video series on FLOSS Linux audio tools: https://www.youtube.com/watch?v=hdPPPeq82hw&list=PLf_MTToSAx...

Specifically, you can play with it yourself with a tool like ZynAddSubFX's (https://zynaddsubfx.sourceforge.io/) virtual keyboard, where you can enable microtonal under scale settings and tweak to whatever tuning you want (see https://www.youtube.com/watch?v=hdPPPeq82hw&t=669s for where the video above starts showing this).

A simpler and shorter video discussing the history of just intonation and equal temperament, with examples: https://www.youtube.com/watch?v=bCYcS57eCqs

You have a good ear, you are hearing the small errors inherent in "twelve-tone equal temperament"[1] which is a compromise between numerical ideals and practical instrument-making.

[1] https://en.wikipedia.org/wiki/12_equal_temperament

Equal temperament is used to allow playing in arbitrary keys without retuning…for example D and E flat are equally in (and out of tune) with equal temperament. With Pythagorean or Just tuning, instruments are tuned to a specific key (or set of related keys).

Which is to say historically, equal temperament is mostly a result of compositional and performance concerns rather than manufacturing limitations.

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