Historically Informed Practice
for Early Keyboard Instruments
Temperaments – Tunings of the scale in which some or all of the concords are made slightly impure in order that few or none will be left distastefully so. Equal temperament, in which the octave is divided into 12 uniform semitones, is the standard Western temperament today except among specialists in Renaissance or Baroque music.
Equal Temperament – Equal temperament is the most mathematically perfect of all temperaments. In it the Pythagorean comma is spread equally over all twelve fifths, flattening each by 1/12 comma or 1.955 cents. Twelve tempered fifths exactly equal seven octaves, wich closes the circle of fifths. This gives a simple and neat mathematical relationship between the intervals, but at the expence of all intervals except the octaves being out of tune. Furthermore the distinction disappears between the fourteen intervals, B and Cb, C# and Db, D# and Eb, etc. In fact the equal tempered notes come nearly midway between the notes C# and Db, and exactly between F# and Gb, and are thus good compromise notes. The other pairs are not so nicely split and one could say that there are no D#, G# or A#s since these are 1/4 semitone out of tune. Equal temperaments was perhaps proposed by Aristoxenus c. 350 B.C. It was certainly used in Spain in 1480 for placing frets on guitars. It was used for tuning lutes from the 16th century. Equal temperamnt was rejected for tuning keyboard instruments for several centuries because the badly out-of-tune thirds and sixts did not fit the contemporary harmony. Equal tuning did not become widespread on the Continent until around 1750, and not for another 100 years or more in England. It is an unrestricted 'good' temperament in which all keys are equally good in intonation, but it is also true that they are equally bad in intonation. This results in a uniform 'key-flavour', in which all keys sound the same except for pitch.
Quarter Comma Mean Tone Temperament – 1/4 comma mean tone temperament was devised to suit the modal music of 400 years ago. It was supposedly widely used for 300 years, although in the later years it probably existed in several modified forms. In this temprament th eight major thirds and eight minor sixths are exactly in tune. Nine minor thirds and nine major sixths are better in tune than in equal temperament. This temperament dates from at least the end of the 15th century. The mean tone system was given a precise mathematical form by G. Zarlino in 1517 and later advocated in Spain by F. Salinas. In this temperament the Syntonic Comma (21.5 cents) is spread over four fifths. Thus for example, ascending the four fifths CG, GD, DA and AE produces a true third CE. This is repeaed around the circle of fifths, and in all, eleven fifths are equally flattened by 1/4 Syntonic Comma, that is by 5.38 cents. This is about 2 1/2 times the amount by which equally tempered fifths are flattened. Thus eight thirds are true at the expence of the fifths. However, if two thirds in the octave are true, the remaining third must be very wide so that three can fit into an octave. Thus all the lesser deisis (41 cents) has to be used up in sharpening this third. This results altogether in four very sharp thirds.
Another problem arises from the flattening of eleven fifths each by as much as 5.38 cents and that is the Pythagorean Comma is not only eliminated, but it is exceeded by over 1 1/2 times. This arises because 11 x 5.38 = 59.18 cents, whereas the Pythagorean Comma is 23.46 cents. Therefore the excess 59.18 - 23.46 is 35.72 cents. This error appears in the very out-of-tune twelfth fifth which is thus over 1/3rd semitone sharp. This is the notoriuos 'Wolf Fifth' presumably so-called because it 'growls' so badly. It was placed at various positions round the circle but most often was placed on the fifth G#Eb. Its position decides which are the good and which are the bad keys. Corresponding to this bad fifth is of course a bad fourth EbG# which is the same amount flat. These gaps had either to be avoided when composing, or else they were exploited for enhanced musical effects.
Another method of overcoming these wolves on old harpsichords and organs was to use the split keys. The 'black' keys G# and Eb were therefore divided in half along their length with the back half raised, so that on one half D# sounded and on the other half Eb sounded, and correspondingly G# and Ab.
circle of fifths
1/4 comma mean tone
copyright HIP f EKI 2005
