Where is the well temperament option?

• Mar 1, 2022 - 13:08
Reported version
S5 - Suggestion


In reply to by Exaltation8

Jojo is not wrong, but it is not the whole story, and I would suggest that the original question is unanswerable. In summary, "Well temperament" refers to a class of temperaments that allow music to be played in a wide range of keys; one such temperament is equal temperament. So to ask "where is the well temperament option?" requires the response "which well temperament do you want?". Or the response "The default is equal temperament which is well tempered".

Much of what follows is based on "How Equal Temperament Ruined Music (and why we should care)" by Ross. W. Duffin - I recommend it as an easy-to-read discussion of how tuning systems have evolved.

Well temperaments try to solve the problem of the "Pythagorean comma". This refers to the impossibility of creating a scale of 12 notes such that octaves all have pitches with frequencies in the ration 1:2 and also fifths which all have pitches with frequencies in the ratio 2:3. We can identify the 12 notes of the chromatic scale with a cycle of 5ths, C-G-D-A-E-B-F#-C#-G#/Ab-Eb-Bb-F and back to C. However, if all those 5ths have their pitches defined as having frequencies in the ration 2:3 the first and last C in the chain should be 7 octaves apart (with frequencies in the ration 1:128). In fact it is about a quarter semitone too high (actually it's 1:129.746 if you do the maths). That difference is known as the Pythagorean comma.

Equal temperament solves the problem by making all the 5ths equal but with a frequency ratio less than 2:3. In fact each 5th is narrower by 1/12th of the Pythagorean comma. It means that notes a semitone apart have frequencies in the ration of 1:2^(1/12) and fifths (seven semitones apart) have frequencies in the ratio 1:2(7/12). Thus instead of having a frequency ratio of 2:3, the fifths in equal temperament have frequency ratio of 2:2.9966.

However, as noted other solutions have been tried. In Bach's time there were several individually different tuning systems in common use including among others, those that were devised by Andreas Werckmeister, Johann Phillip Kirnberger, Johann Georg Neidhart, and Francesc'Antonio Vallotti. These distributed the Pythagorean comma by applying non-equal adjustment of the fifths. This resulted in keys that differed in "flavour" but which were all usable. And all of these were referred to as "good temperaments" or "well temperaments". One example is shown in this figure (from Duffin's book)


In fact there is good evidence that the temperament in this figure is the one that Bach used and therefore is the temperament he had in mind when he wrote his "Well-tempered clavier" preludes and fugues. But note that this is not equal temperament.

Interestingly, Sebastian Bach's son, Carl Phillip Emanuel seems to have favoured equal temperament and this has tended to create a false assumption that he was following his father's preference.

In reply to by SteveBlower

Well thank you so much for detailed information on tuning systems. I believe that there are more than one well temperaments and it's kind of a category in itself. What are your thoughts on Lehman's Bach tuning?

It is challenging to find one absolute tuning system that is perfect in purity and yet practically usable across all keys and modulations. Equal temperament does the job well, but do you think that we have a better alternative in any of the known methods? Your personal thoughts please.

In just intonation, it is said that the fifths and major thirds are pure (correct me if you must). I am to curious about -- are other intervals pure too? assuming that the root note is C, for example and we tune for JI w.r.t. C, what about the C Minor third?

In reply to by SteveBlower

Actually, the term "well temperament" is an invention of the American piano tuner Owen Jorgenson. Historically, it was never used because "well" (wohl in German) is an adverb, not an adjective. Werckmeister was the first to speak of a "well tempered" keyboard instrument, but when speaking of the temperaments themselves, he used the adjectives "good" (gut) and "correct" (richtig), and always in comparison to what he called the "false" (falsch) temperament, 1/4 meantone. His classification of "well-tempered" systems most definitely included Equal, which he referred to several times, most notably in the 1707 Musicalische Paradoxal Discourse, in which he flat out stated that if one were to tune ET on a keyboard, a "well-tempered harmony" would be the result. So the idea that it specifically refers to unequal systems is part of the Jorgenson modern myth.

These systems are properly called "rational circulating", and the primary authors are Werckmeister, Neidhardt and Sorge. Neither Kirnberger nor Vallotti are part of this tradition. Furthermore, the primary goal is not so much resolve the Pythagorean comma as it is to create a circulating system in which all keys are more or less usable. Neidhardt described the methodology; the sizes of all 12 major thirds are first chosen, then the first three 5ths (C-G, G-D, and D-A), and then all else results by mathematical necessity (Sectio Canonis Harmonici, 1724, p.20). So while the P. comma is indeed resolved, it is but a secondary goal. The resolution of the lessor diesis among the four stacks of major thirds on C, G, D and A is the primary concern.