Source: Spiegel der Orgelmacher und Organisten (Mainz: 1511)
Arnaut Schlick’s work is universally regarded as being an extremely important document, not only as the earliest text dealing with the design and construction of organs in detail, but also as the first complete description of a tempered tuning. Nonetheless, his temperament has been all but completely ignored by the modern HIPP movement. The reasons for this are various: (1) a wide-spread lack of recognition of the importance of Mollidfied Meantones in general and of those of German authors in particular; (2) a common misperception that Schlick’s instructions are too vague to allow a reasonably-precise determination of the structure of the temperament; and (3) a veritable plethora of modern interpretations, the primary difference among them being whether or not the temperament is considered to be “circulating” or not. Some modern authors prefer to view his temperament as a “Germanic proto-meantone”, i.e a first step in the process of moving away from Pythagorean toward the eventual pure thirds of a “fully-formed” 1/4 comma regular meantone. However, regardless of how we conceive of it, this temperament essentially deals with the same problems and uses the same solutions as those of the later authors.
Like almost all Mollified Meantones, Schlick uses a vague language which is open to interpretation. However, due to the logical constraints of the various clues he provides, the range of possibilites is not that large. The conclusions I have used for the two variations provided here are given below.
Disclaimer: the attribution of comma fractions to the tempered fifths is done by an automatic table look-up function. If the exact value is not in the table, the system grabs the nearest lower value. Naturally, my table does not have all possible fractions of both commas for both narrow and wide fifths. Therefore, with all modified meantone temperaments, for those fifths which “close the circle”, i.e. fill in the gap left by the departure from pure meantone logic, the amount of tempering has been somewhat arbitrarily chosen, exactly as one does when tuning an instrument, although in this case, working with decimal values for the proportion of the fifth. Therefore, the final size may not represent any fraction of any given comma, and there is no guarantee that the amount of tempering indicated as a comma fraction in the graphic is precise. The value indicated in cents, however, is derived from the proportion, and is therefore completely trustworthy.
Schlick’s instructions paraphrased
Reference note: Bass F [4’ register?]
F<-c: Hovering somewhat lower, as much as the ear can stand
c<-g: Also like this
d1<-d: pure and good
d<-a: hover on the low side
e1<-e: completely pure
e<-b: also on the low side as previously said.
Tune the upper and lower octaves of all these notes, and then you have all the naturals.
Comments about the results of the above: Octaves are to be absolutely pure. The fifths must be small so that the major thirds do not become “too strong and too high”. While the thirds are not perfect, being somewhat too high to be pure, special attention should be given to the central thirds CE, FA, and GB, making them better than others, because they are more often used. However, the better they are, the worse G# is in relation to E and B, but that is not too important, for reasons to be given later.
Tenor f<-Bb: not a good fifth, but hovering on the high side, as much as possible for the third Bb-d.
Bb<-Eb: also on the high side relative to Bb as is said above about the adjacent fifths.
D#<-G#: not on the high side, like the others, but lower than the fifth desires. This helps E and B in cadence to A, but it is not a good third, and one can hide the “hardness” of the third by making a trill, a break, a little run, etc, as every organist knows how to do in such cases. Some say that the note should be made more of a real G# than an Ab, but that “weakens the music and takes away its proper characteristics”, and those who previously thought this way have since come around.
Starting from B
B<-f#: hovering weakly on the low side, so that the third with D and A is useful and not too high, since the cadence to G is often used, while the fifth above B natural is seldom employed.
f#<-c#1 [exact octave location not discernible]: tuned rather so that it is useable with AE in cadence to D (see discussion below). It will be too low against G#, but this is not important unless one wants to go “through all the semitones by musica ficta”, which, although not done, is something composers might want to “play with” out of inventiveness or curiosity.
Summary and comments
It is clear that all fifths are narrow with the exception of two, the Wolf fifth C#-G# and the fifth G#-D#(Eb), which is intentional tempered wider than pure. In other words, there is an unbroken chain of narrow fifths from Eb to C#.
Although it appears as though not all of the narrow fifths are to be tempered by the same amount, they are nonetheless all tempered specifically in order to make the resulting major thirds (and in some cases, perhaps minor thirds) “good”. It is clear from in the opening comments (not summarized here) that a Pythagorean ditone is “too high” or “too hard”, and therefore a “good” third is one which is noticeably narrower than the ditone. Nonetheless, such thirds are also specifically defined as not being absolutely pure. While the exact degree of impurity of a “good” third is not stated, and even though the accumulated tempering is partially alleviated by the one wide fifth and the four slightly less-narrow fifths, it must nonetheless be pure enough so as to cause the Wolf fifth C#-G# to be noticeably too wide. In other words, the descent from Eb down to C# caused by the chain of narrow fifths must be steep enough so that the drop in elevation cannot be recuperated easily and gradually by having all of the remaining fifths tolerable, even though the accent is achieved in two steps, one of which is relative small. At the other end of the range of possibilities, the central descent cannot be so steep as to result in pure major thirds.
These restrictions limit the possibilities for the tempering of the central chain of fifths to between 3 and 4 cents too narrow, which produce central major thirds which are either 1/2 or 1/4 Syntonic comma too wide, respectively. This provides a subtle though not insignificant variety of possible third quality, since the worst case scenario (fifths only 3 cents narrow) produces thirds which beat almost twice as fast as the best case. Considering that Schlick specifies that the tuning is to be done very low on the keyboard (though most likely with a 4’ register, as he says to tune the “klein werk der orgell”), it is obvious that the primary evaluation is to be done by listening to the qualities of the thirds, since the fifths will only produce a change in their sound at a very slow rate, around 1/3 to 1/2 “beats” per second (i.e. 3 to 2 second for one cycle of change).
Additionally, there is a strong implication that the central chain of fifths from F to B should be somewhat narrower than the others in order to make the thirds F-A, C-E and G-B better than the other good thirds. However, the others must yet remain good enough to work well in the stated cadencial progressions, with the implication that they have no need for the sort of disguising trickery recommended for using Ab as a G#. This argues more in favor of central fifths rather narrow than closer than to pure, for if the other fifths are to be slightly purer to make the other thirds slightly worse, if the central fifths are too close to pure, one runs the risk of reducing the width of the “Wolf” fifth so much that it no longer retains its lupine quality.