# Where is the well temperament option?

• Mar 1, 2022 - 13:08
Reported version
3.6
Type
Functional
Frequency
Once
Severity
S5 - Suggestion
Reproducibility
Always
Status
active
Regression
No
Workaround
No

That's what MuseScore uses by default, no plugin needed

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.