This translation is a work in progress; eventually I hope to provide the complete text. I would like to thank Thomas Dent for providing me with the German transcription and for making many suggestions about the translation. Bill Jurgenson has also helped clarify some points.
Copyright Paul Poletti 2007
In this little treatise [on continuo playing] a tempered keyboard instrument has been mentioned several times. Because a beginner often doesn't know what temperament is and what's involved in tuning a keyboard instrument, I am prompted to briefly illustrate how one can tune a keyboard in a tempered manner.
Temperament has its origin in the fact that when a keyboard is used, it is not possible to have all the consonances pure if one moves from one chord to another. Therefore, a little must be given to the one consonance and a little taken away from the other in such a way that a pleasing temperament is formed. When one of the tones of a consonance stands a little too high or too low against the other, this is called "hovering" [see note 1]. This term originated primarily among organ builders, in that when they are tuning two pipes together and they are almost pure [against one another], such pipes, when they are sounding together, make a trembling or shivering sound. The closer they are to being in tune with one another, the slower the trembling, and when they are finally tuned together, the trembling or shaking is no longer heard, and the two pipes sound as one. It is this trembling or shaking which the organ builders call "hovering". It also has a double meaning, such as when the upper note is too high against the other, one says it is hovering on the high side, and if it is too low, in that case one says it is hovering on the low side, or the reverse. If the upper note is too high against the lower, one says that the lower [note] is hovering on the low side, and in this sense the tail can be attached to the dog or the dog to the tail. These terms often give rise to confusion, and the one person can't understand the other, whereby much strife and conflict between musicians is brought about, which is another reason why I mention this.
However, on string instruments like spinets, clavichords, and such, where this hovering or trembling is not so easy to comprehend, and a student who can temper an organ fails [at this? - manuscript unclear],the best way to experiment is with a good, stable regal; thereby one can become practiced and secure with such temperaments. Some experience is required in this, for even when one already knows that one note of a consonance should hover by one-, two-, or three-quarters of a comma against the other, our poor sense of hearing cannot judge accurately if the [degree of] hovering is too great or too small, or if they beat too slowly or too rapidly. There is also a big difference when two small or two large pipes are tuned with one another. For example, I have to tune an octave, such as c and c', 4 foot and 2 foot, or c' and c", 2 foot and 1 foot [see note 2]. If c-c' were a comma too small, or impure, and if c' and c" were to have the same distance between them and also be a comma too small or impure, the c and c' between the 4 and 2 foot would tremble, or hover, once again more slowly [i.e. half as much] than the c' and c", and so on. The fifths, however, will tremble faster or slower according to their proportion of 2/3 when the deeper and higher pipes, or larger and smaller, are sounded or tuned against one another [see note 3] That's why it is difficult to determine with complete precision these small differences with the sense of hearing alone. Meanwhile, by means of a monochord demonstration, one gains the certainty and guidance of how the ear can tackle the problem, in that with our monochord, the different manners of tempering and tuning can be found out. However, here we want to proceed in a simple and mechanical way, since the beginner, as well as many others, don't know what a musical comma is, or as the Dutchman Doctor J. A. Ban [in his] Zangle-Bloemzel has called it, a snippet [see note 4]. Certainly all the fifths could be tempered so that the upper [note] is hovering 1/12 of a comma downwards against the lower [note], and proceeding in this manner the major thirds become too large by 2/3 [comma] and the minor [thirds] too small by 3/4 [comma], all of which one would endure if one wanted to go through the entire instrument and play all pieces in all keys. However, in our instruction we don't want to think about any commas, and as stated, merely proceed in a simple way, and describe the procedure so that all the diatonic and chromatic intervals which are the most often used these days retain the most purity.
He who wants to tune the 12 keys of the octave in this manner and put the whole keyboard in order can take at will [tenor] c (4 foot tone) [see note 2] as the fundamental key and tune it at whatever pitch level (be it Choir Pitch or Chamber Pitch) he desires. To this he may tune [middle] c' pure. From the c, one sets g a fifth above so that it hovers a little bit low against the c. To this g one can set d' which again hovers a little bit low against the g. To this d' one tunes the d [an octave down] absolutely pure; to this d one takes again the fifth a above, letting it hover just a little bit on the low side. To this a, the fifth e' is again tuned so that it also hovers just a little bit on the low side. Now one holds [plays] this e' with the c or the c'; if this third c-e' or c'-e' is tolerable, that is, if the the e' is not much too high, the process has been successful. This is the first test, since all the major thirds must hover on the high side (against the lower of the two notes). However, if this e' is much too pungent or too high, all the fifths must be corrected a little and lowered until the e' hovers by a tolerable amount on the high side. Once this e' is properly set, one can continue and tune the e an octave below completely pure. To this e one tunes the h [so that it is] once again hovering a very subtle amount on the low side. At this point, one can try the major third g-h, in which case the h must also hover a little bit high against the g, as much as the ear can bear. One can also directly test g-h-d', for if the triads sound good, the thirds will all be bearable. Thus the major third g-h is the next test. To this h one again sets the fifth above, h-f#', in such a way that the f#' again hovers a very small amount on the low side. To this f#' the f# [an octave below] is once again tuned absolutely pure. Thereafter the d-f# or d'-f#' can again be used as a test, and the f# or f#' must once again hover on the high side against the d. To the f# one tunes the c#' as a fifth which must hover below itself to the same degree. From this c#' as a test one takes the major third a, in which case the upper note must hover on the high side against the lower (as is the case with all major thirds). To this c#' one again tunes the c# a pure octave down; to this c# the fifth g# can be tuned almost pure; the test for the g# is the e. This third will tend to sound a bit strident, but if one imagines using the g# in place of ab, as in f-ab-c', it can't be any other way. To the g# the fifth d#' is tuned. This d#' can hover a little bit above itself taken with the g# [wider than pure!] in order that it makes a tolerable major third to the h and [as an eb] to the g'. From d#' the d# an octave below is once again tuned pure. From this d# the fifth b is tuned, which can also hover above itself a very little bit [wider than pure] so that the corresponding third with d' becomes tolerable. From the b the fifth f' is taken, which either hovers a little above itself once again, or is completely pure, accordingly to whether the f can be held with the c', or as the last test, the f with the a as a major third. First, however, the f must be tuned pure with the f', after which it can be tried with the c' and the a. Now should one or another of the fifths prove to have been tuned a bit too low or too high, one can correct them all, and the thirds will all be tolerable, in that they can bear much more [tempering] than the fifths, since they are not so perfect according to their [mathematical] proportions.
In this way, all the simple consonances and several of the compound ones are now pure. From the c c# d d# e f f# g g# a b h c' c#' d' d# e' f' f#' all the other notes can be tuned by pure octaves, the upper as well as the lower.
[as yet untranslated remarks primarily relating to clavichord tuning]
I never cease to be amazed when someone asserts the old hypothesis that all of the fifths should be set low by 1/4 comma for the entire keyboard and all the thirds should be pure, for I have found that in their organs most of the thirds are too large and hover above themselves, which is as it must be, and, practically speaking, cannot be otherwise. They obviously tune in part according to habit, without reason, and it simply turns out as it does, because they can't fathom if the thirds are hovering or not because they can't actually be made to tremble [see note 5]. Fifths which are hovering by 1/4 comma, as prescribed, are also not found in the old organs. The outcome would be curious and the last fifth would only be fit for tossing to the dogs or the ravens. These fifths which are 1/4 comma too small, especially when they are played alone without the assistance of the thirds and if they are tuned a little bit too low, cause a dissonance which is so hideous and lame that no healthy ear can possibly approve of it. A third which is 2/3 to 3/4 [comma] too large sounds much more pleasant to the ear than such a false fifth, in that the more perfect the consonance, the less [tempering] it can bear. In that a third is less perfect than a fifth, it follows that the octave can suffer the least amount of tempering, that is, absolutely none, as experience itself demonstrates when the proper checks are employed, according to true proportions. While almost all of the fifths in this explanation are hovering a bit on the low side, that is, they are too small, by no means is it as much as 1/4 comma, but only by about 1/8 to 1/12 comma, so that no fifth suffers so much that it would annoy the ear [see note 6].
[as yet untranslated closing remarks about old organs being tuned in meantone]
"Hovering" is in fact a perfect metaphor for the act of setting a tempered interval. If we think of one note of an interval having a right and proper elevation above or below the other note, that is, the position at which it creates a pure interval with the other note, any slight deviation from this position can be seen as a sort of "hovering" slightly above or slightly below this proper position.