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In addition, the article puts forth a commonly held (but false) notion that equal temperament destroys any sense of key color or identity, apart from pitch. That would be true if every instrument or voice sounded exactly the same throughout its entire range, but that is never the case.

Even in equal temperament a G minor chord sounds different than a D minor chord because they fall on different ranges of the instruments and the human auditory system. If you try transposing a piece in G minor to D minor, you'll notice a clear difference. You could either go up a 5th or down a 4th. If you choose to go up a 5th, things will sound quite a bit brighter. Conversely, going down a 4th could very well cause the piece to sound muddier, depending on the orchestration. At any rate, the other forms of tuning have other effects, of course, and are worthwhile to explore.

> brighter

Or shrill.

> muddier

Or warmer.

(Just a nit, but I think it's important to present these qualities in both their positive and negative forms.)

Yeah, I had to laugh at that, too. I remember in my music theory class, one time the professor played something in D-minor, and someone in the class pulled out that line from Spinal Tap, "D minor is the saddest of keys." We chuckled, but the prof said, "Yeah, it really is!" and we had an interesting discussion about it.

> In this way, the Well temperament can be seen as an approximation of equal temperament in a time when the mathematical tools to achieve it were not really known.

Is there any evidence for this narrative? Like anyone from the time writing about this who said "the keys are not all identical, but this well-tempered tuning is the best we are capable of doing."

> The purpose of Well temperament, used on the harpsichord, was to get rid of those limitations and gain the ability to transpose everything freely while remaining consonant and in-tune.

It was certainly designed to make all keys usable, but it doesn't necessarily follow that all keys were intended to be identical.

In fact, equal temperament as a concept was known and discussed by theoreticians since at least the 16th century, and was advocated as an ideal solution for tuning and adjusting frets on fretted instruments (namely the lute and the viola da gamba). But equal temperament stayed more or less a theoretical concept, though players of fretted instruments apparently employed an approximation of equal temperament.

Actually, it is important to note that the problem of temperament is generally related to keyboard instruments, which have a fixed intonation, and 12 notes to the octave. Not so on wind or string instruments.

> I like natural intervals, but dividing the octave in 12 tones is a rather inappropriate way of using them.

This is a gross misunderstanding of how we came to have 12 notes to the octave. Actually, depending on how far back you wanna go, we started with 6 notes (the hexachordal system), ut (C) to la (A), to which were added the mi and the fa of the hard and the soft hexachords, which became B and B flat. The rest of the black keys on the keyboard came from further modulating the hexachords.

In the 16th century, there were actually numerous keyboard instruments built with more than 12 notes to the octave, reaching 19 notes and sometimes even 31 notes to the octave. Having separate keys for e flat and d sharp meant being able keep using meantone tuning while playing in such remote keys as b major and d flat major.

> It's not the ultimate system nor the end of temperaments, but it's been really, really useful.

That's true, it's a practical system which makes instruments more approachable and makes it easy to play together, but there are downsides: ET removes a lot of the charm of harmony (all keys sound the same and all chords are more or less discordant). Equal-tempered thirds are actually very discordant (that is, out of tune), but people have become so used to ET, that you play them something in meantone and they'll claim the pure thirds are out of tune!

> To me, it's pretty clear that the WTC is Bach gleefully exploring the freedom of transposition-invariance and free modulation. In other words: Bach would have loved the equal temperament, and this article would likely give him a hearty chuckle.

While Bach was known to have been critical of mean-tone tuning, he was, according to evidence, a practical guy who employed the practices of his time.

Yes, one can imagine that he would have preferred equal temperament to well-temperament. One can also argue, and many do, that Bach would have preferred the piano to the harpsichord. Maybe he would have also preferred a Yamaha D7 to a pipe organ. One can imagine...

I have a relatively good understanding of how we came to have 12 notes to the octave, you just misinterpret my statement: keyboards already had 12 notes per octave long before ET came into widespread use. And trying to reconcile this division with a tuning system that relies on natural intervals is an awkward exercise, this is the inappropriateness that I speak of.

As for the discordance of equal-tempered thirds against the perfection of pure thirds, this may be true in a mathematical sense, but not necessarily in a human sense. Our brains are logarithmic in many regards, and a lot of research argues that a logarithmic scale is something for which we have affinity.

Here are a few starters to go through the recent evolution of this debate:

Just Intonation Confuted, J. Murray Barbour, 1938, and all the subsequent works that cite it

Violin intonation, PC Greene, 1937, and the many works that cite it