# On Tuning: What's in a Temperament? (Well-Temperament vs. Equal Temperament)

I just had my whole world shattered this morning when I was unwise enough to do a "quick" Google search on what exactly "equal temperament" means before I had started my work. Of course, the loudest voices in such a search all scream "IT'S FREAKING WRONG, YOU HEATHENS."

Is it acceptable to tune an instrument to well-temperament in this day and age? If so, what are the exact frequencies to tune to? Will it require, on an intonation-based instrument, such as the violin, a re-learning of what each pitch should be (Will I be constantly trying to flip back into equal temperament?)Basically, is there a temperament system which isn't totally equal like what we use today that can be used in practically any key without "wolf" chords?

(On the side, when a person says "concert pitch," I suppose that means he's referring to equal temperament, right? Wrong? Ahdunno, that's the job of the enlightened geniuses about to answer who will undoubtedly storm this question and tell me every place where I got my facts wrong here. But, whatever, that's what I want...)

You can approach the subject of temperament either as a mathematical exercise, or as a practical one (i.e. what it sounds like).

Mathematically, you are trying to find the "best" solution to a problem that is insoluble. Empirically, the intervals of a perfect fifth and an octave sounds "exactly in tune" if the frequency ratio of the notes is 3:2 and 2:1. The problem is that you can't make an exact octave by stacking up exact fifths. If you build a cycle of fiths like C, G, D, A, B, ... you eventually get to an interval C - B sharp with a frequency ratio of 531441:262144, which isn't quite the same as the octave C - C with a ratio of 524288:262144 or 2:1.

There have been almost as many different ways to "make B sharp the same note as C" as the number of people who have written anything about the problem.

The current "standard" for western music is equal temperament, where you split the octave into 12 equal parts. That makes a "not quite perfect fifth" with a ratio of 2.9966142 : 2, instead of 3:2. That doesn't look much different on paper, but it is just about audible.

There were several different solutions to the problem in use in the 18th and 19th centuries, all of which allowed playing in every major and minor key. These "well-temperaments" result in semitones of slightly different sizes, and therefore each of the 24 keys has a slightly different sound - and 18th and 19th century composers knew about that, and used it intentionally.

This is a good demonstration - Bach's organ prelude and fugue in D (BWV 532). The start is basically D major scales and chords, ending with an A major chord. At around 30 seconds this is suddenly followed by passage built around an F# major chord, and the tuning of the F# chord is noticeably different from the D and the A - probably a "WTF???" moment for Bach's original listeners, who were not used to this sort of thing.

There are several unequal well-temperaments that are currently used for "historically informed" music performances. https://en.wikipedia.org/wiki/Well_temperament lists the common ones, and has links to the details of most of them.

Unlike a keyboard instrument, the violin is not restricted to playing in any "fixed" temperament, except for the tuning of the open strings. In practice, individual notes are "tempered" according to the musical context, often by much more than the difference between a well-temperament and equal temperament. I've seen claims that professional players sometimes pitch-shift individual notes by almost a quarter-tone.

In contemporary music (since about 1950) most of the interest has been in equal-tempered tunings but with more than 12 divisions per octave. By dividing an octave into more than 12 equal parts, you can get intervals which are "closer to perfectly in tune" than Equal temperament, but still "play in any key". 19, 41 and 53 divisions are cases that work out well and have been explored musically.

• Given that every scale has two (or more?) kind of step, why would anyone expect to be able to build a scale from only one? The problem as I see it is that you can't meet three desiderata (octave, fifth, third) with only two kinds of step. Oct 25 '16 at 22:55
• @AntonSherwood "why would anyone expect to be able to build a scale from only one?": to be able to modulate. Unequal temperaments are lovely in certain keys, but of course awful in others. Much 19th century music would sound bad in unequal temperaments (even a lot of Baroque music is unlistenable in mean tone temperament, which was for the most part superseded by Werckmeister and others before the end of the Baroque). Nov 6 '19 at 15:49
• @phoog You've answered a different question: why one would want to attempt it. Nov 6 '19 at 16:06
• @AntonSherwood fair enough. But experience shows that people not only attempt it but succeed. That the major third of 12-tone equal temperament is so far off from the acoustical pitch does not prevent scales built from 12-tone equal temperament from being scales. Nov 6 '19 at 16:30

Violins are tuned by tuning the A and then tuning all other strings in perfect fifths in relation to A. The idea of Werckmeister III tuning, the most commonly employed well-tempered tuning, is actually pretty compatible with that scheme, namely using 8 perfect tuned fifths and then offset the accumulated +16cents of tuning error (a perfect fifth is 2 cents above an equal-tempered one) then over four non-perfect fifths. Unfortunately, those fifths are C-G, G-D, D-A and B-F♯, so a Werckmeister III G is 8 cents above a normally tuned violin G (and you'd be even off 12 cents for a viola C).

By the way, Bach wrote for "Well-tempered Clavier" (likely as opposed to mean-tone temperament common for keyboard instruments at the time, mean-tone being based on pure and unpure thirds rather than pure and unpure fifths and thus making unusual keys much less playable). He did not invent it, and it is not clear just what well-tempered temperament (of which there are several different ones) he had in mind for that work cycle.

• Violinists (and other string players) playing with historically tempered keyboards actually tend to tune each string to the keyboard, since as you note failing to do so can leave the open strings fairly far from the corresponding notes on the keyboard. Two of the four small fifths you note are open strings in violins, violas, and cellos (G-D and D-A), and one in violas and cellos only (C-G). Nov 6 '19 at 16:38
• Some say the intended temperament for the Well-Tempered Clavier is indicated on its title page: rolf-musicblog.net/what-tunings-did-bach-use Aug 2 '21 at 10:05

The term "well tempered" is generally used for what you refer to in the end: a temperament that is not-equal, but is (reasonably) playable in any key. The Werkmeister tunings are the most well known of this sort. It's my impression that many pipe organs are still tuned in this manner (so that is the context you'll often hear it in), due to way that the sustained nature of the notes works with the common repertoire.

(Disclaimer I'm not a violinist) In terms of violin performance I've heard the following: if you tune your violin using harmonics just by ear so that they form perfect (consonant) fifths, so that G->D is a perfect 5th, D->A is a perfect fifth and so on. The interval between the G and E string is a Pythagorean sixth, which is noticeably off (to some people) from a (5-limit) just intonated 6th and/or an equal tempered major 6th. So some violinists "temper their fifths" when tuning by ear, though by no means is this universally accepted as ideal, especially not in all situations, e.g. playing with a fixed pitch instrument.

For the most part, temperament is a problem for fixed pitch instruments. On a violin, for any fingered note, you can and probably do adjust the exact position of your finger to effect an intonation.

Just a quick note (hehe) to your on-the-side question about "concert pitch". This does not refer to temperament but rather to the overall pitch level, usually based on how high or low the A is. Current concert pitch is somewhat standardized at A=440 Hz, but here in Vienna, for instance, it's more like 444.

• Thanks for that. I was gonna edit it out because it's unrelated to the main point, but then your wonderful descriptive answer won't be right. So I'll just leave it there, to magnify my musical illiteracy. Which is getting smaller. ¯\_(ツ)_/¯ Oct 25 '16 at 13:19
• Hey, no problem. Being illiterate in countless fields, I share your shame, lol. Cheers, Scott Oct 25 '16 at 13:32