The Septenarius: Werckmeister's mythical tuning, made reality?

Andreas Werckmeister (b. Benneckenstein, Thuringia, 1645; d. Halberstadt, 1706) is known as the first advocate of circular irregular tuning, in which the accidentals ('black notes') can take a dual role as either flat or sharp, and all intervals should be acceptably tuned, though without any key having perfect intonation. Through some accident of history one of his organ tunings, listed as No.III in his treatise Musicalische Temperatur of 1691, has become much the most well-known — although he presented four different "good temperaments", as he called them, in that work, as well as giving other advice and tuning instructions later in life.

In the case of No.IV, the neglect may be explained by the fact that some commonly used fifths are particularly flat, while keys with many flats or sharps are, relatively, quite mistuned. Werckmeister designated this tuning primarily for "diatonic" playing, meaning that one should stick to relatively simple keys. But if that is the case, it hardly offers much musical benefit over the meantone tuning which was the default in Werckmeister's day.

The last tuning presented, the "Septenarius", has been not so much neglected, as bizarrely and completely misrepresented. That is why I have called it "mythical", since most of the versions that have appeared in supposedly historical surveys have simply failed to present this tuning as it appears in Werckmeister's text — although what he says is clear enough. Various authors have called it impractical, numerological, purely speculative; some have falsely represented the tuning, in an apparent effort to fit it into the mould of the other three "good temperaments".

First, here is how Werckmeister introduced his temperaments in 1691. Explanations in square brackets and emphasis are due to me.

(Chapter 20) ... we want to display on the monochord the two temperaments that were put forward in the Orgelprobe [of 1681] (...) we have numbered each one separately on the copper plate [illustration], as
No. 1 the pure scale [just intonation], in which one can have all pure consonances from one tone to another, with the addition of the subsemitones [multiple pitches for a single note] from which it can be seen how far one consonance or dissonance differs from another and how the temperaments can be made;
No. 2 is the wrong temperament where all fifths are all fifths are tempered by 1/4 comma.
No. 3 is a correct temperament, which is likewise divided up through quarters of a comma, thus some fifths are pure, some tempered 1/4 comma upwards [this seems to be a mistake – No. 3 contains no wide tempered fifths!], some tempered downwards.
No. 4 is a temperament also included in the Orgelprobe and divided up through 1/3 comma.
No. 5 is another additional temperament divided up through 1/4 comma. After this
No. 6 is an additional temperament which has nothing at all to do with the divisions of the comma, nevertheless in practice so correct that one can be really satisfied with it.

This already contradicts the claim that the No. 6 tuning was, for Werckmeister, merely mathematics or symbolism. (It also gives the lie to a chain of authors – starting with Dupont and Barbour – who have tried to fabricate a temperament divided up by 1/7 comma out of the Septenarius.) In the 17th century, there was really no clear division between mathematics and music: Werckmeister's publications often mix what appear to be purely theoretical and practical topics.

The monochord

The direct link between mathematics and musical notes was provided by the monochord, a simple instrument consisting of a wooden board with a length scale carefully marked out along it, and one or more wires or strings stretched along its length between small "bridges". By moving one or more bridges one could obtain different sounding lengths of string while keeping the tension constant. It was found very early in history that simple ratios of string length produced easily recognizable musical intervals: the octave arose from a ratio 2:1, the fifth from a ratio 3:2, the major third from 5:4 and so on. By constructing string lengths on the monochord one could study the properties of musical intervals with (reasonable) accuracy and at leisure.

For example, one can show simply by looking at the length scale that two perfect fifths and two perfect fourths tuned in succession (3/2 × 3/2 × 3/4 × 3/4) produced a third with a ratio 81:64, which is different from the pure third 5:4 by the "syntonic comma" 81:80. This means that at least one of these intervals must be out of tune on a keyboard, if it has only one key for each named note (C,G,D,A,E etc.). The image above shows a small part of the monochord, with the left-hand fixed bridge and the beginning of the length scale. The two lines marked for the notes D and E give lengths (measured to the right-hand fixed bridge) which differ by the same 81:80 ratio.

Most of Werckmeister's tunings involved dividing the comma into small pieces and distributing the pieces among several tempered fifths, so that none of them was very much out of tune, and the commonly used thirds were also not too impure. However, in the Septenarius Werckmeister bypassed this chopping up of the comma and constructed the ratios of intervals directly by using simple integers. Through a clever choice of 196 = 2 × 2 × 7 × 7 for the fundamental number (representing tenor C) he was able to produce fifths and thirds that were not too far from the pure ratios.

The Septenarius in detail

Here is the way the tuning was introduced:

(Chapter 27) Now we will take up the case of our temperament which arises from the Septenarium; this is very easy to set in the instrument [Werck] and can be achieved in two ways.

Note that the German name for B flat, 'B', and B natural, 'H' are used. Thus flat signs are not needed. The dagger over the string length for D denotes my belief that 176 was a mistake for 175, which gives a scale which is both musically much better, and more consistent with Werckmeister's principles of temperament, as I will explain. (Kristian Wegscheider and Mark Lindley have also noticed this point.)

The "two ways" refers to a variant of this tuning, where the total monochord length was divided into 147 units; in effect F of the Septenarius above becomes C of the variant tuning, and all the intervals tempered and pure are transposed up a fifth. In this variant A is given the number 88 (half of 176) which I believe just as erroneous as D=175 in the table above.

Fifths

In the main text Werckmeister gave a table of the fifths in the tuning, and an explanation to show which were pure and which tempered, and by how much. Unfortunately this contained several errors, for example some narrow fifths are wrongly signalled as wide. I have constructed a (hopefully) correct version below.

Here, '—' means that the fifth is pure; '-x:y' means that the fifth is tempered narrow by the small interval given by the fraction; analogously, '+x:y' denotes a fifth tempered wide by a certain small interval.

What do all these numbers mean? First note that the ratios of tempering are all superparticular: that is, they all have the form y+1:y. So the larger the numbers, the closer the ratio is to unity, and the smaller the interval — thus the less tempering the fifth bears. Now recall that the syntonic comma is 81:80. So a tempering of half a comma would be 161:160 or 162:161; a third of a comma, about 242:241, and so on. For quarter comma, the numbers would be around 320; fifth comma, around 400; sixth comma, around 480.

Now all the numbers here are above 240, except for the G-D fifth of the D=176 tuning, which is about 3/5 of a comma narrow. Such a very narrow (flat) fifth is totally inconsistent with Werckmeister's other writings and temperaments; for example in 1697 he allowed that a fifth might be tempered by 1/3 or 1/4 comma, or less, but to take a whole comma, or 1/2 a comma, was mere ignorance.

After the discussion of tempered fifths, Werckmeister gives the comment:

If these fifths are accurately taken in the temperament, then already all thirds will be found quite bearable and listenable. That in all temperaments the major thirds must be sharp is proved as follows, if C-E and E-G# were to remain pure, then G#-C could not be used as a third, since the excess is about a whole diesis [nearly two commas] too large: but if E-C is sharpened by 2/4 comma and also E-G# raised by 2/4, then there does not remain a comma from G# to C [i.e. G#-C is (barely) less than a comma out of tune], which sounds more bearable...

Thirds

The tempering of major thirds is problematic in the D=176 tuning. Werckmeister constantly emphasized that one should not have pure thirds (interval ratio 5:4) because of the fact that three major thirds must add up to one octave. But three pure major thirds fall short of an octave by the ratio 128:125, the 'diesis', which is almost two commas. Therefore he wanted the thirds to be wider than pure, so that the diesis would be distributed between them and none would be unbearably out-of-tune.

It is also clear that he intended the least out-of-tune thirds (i.e. those least altered from 5:4) to be in the commonly used keys with few sharps or flats; in his other tunings of 1691 this is the case, and he allowed some major thirds that were seldom used to be as sharp as one whole comma above pure 5:4.

With this in mind let us examine the major thirds. In this table they are grouped according to which thirds must add up to an octave, and must share the diesis between them.

Tempering of major thirds, for both D=176 and D=175

Major third	C-E	E-G#	G#-C	G-H	H-D#	D#-G	D-F#	F#-B	B-D	A-C#	C#-F	F-A
Ratio for D=176	49:39	39:31	62:49	131:104	208:165	165:131	176:139	139:110	5:4	39:31	62:49	49:39
Diesis fraction (approx.)	0.2	0.3	0.5	0.3	0.4	0.3	0.5	0.5	0	0.3	0.5	0.2
Ratio for D=175	49:39	39:31	62:49	131:104	208:165	165:131	175:139	139:110	44:35	39:31	62:49	49:39
Diesis fraction (approx.)	0.2	0.3	0.5	0.3	0.4	0.3	0.3	0.5	0.2	0.3	0.5	0.2

Now in the D=176 tuning, the third D-F# is over a comma sharp (0.54 diesis) which would make it the worst third out of D-F#, F#-B♭, B♭-D — despite it being probably the most often used of the three. In fact D-F# would be the worst third of the entire tuning! B♭-D would also be a pure third, in contradiction with Werckmeister's whole idea of tempering.

Apart from these oddities the major thirds are perfectly consistent with Werckmeister's other tunings: the purest lie at F-A and C-E, while the little-used thirds at F#-B♭, C#-F and G#-C are all nearly a comma sharp. The note D#, which must also do duty as E♭, is placed almost equally between B (natural) and G, approaching Equal Temperament in this triple of major thirds.

In the D=175 version we find that the distribution of the diesis is more consistent with the ideal expressed by Werckmeister of favouring commonly used intervals and having no third pure, or more than a comma out of tune. The thirds are also compatible with a transposition of the temperament up a fifth, as in the variant mentioned at the beginning with C=147. This simply means that the purest thirds would be C-E and G-B rather than F-A and C-E. Note that E♭-G in this variant would be almost a comma wide, while the thirds D-F#, F#-B♭, B♭-D would now be the near-equally tempered ones: thus, this tuning would give sharp keys notably better intonation than flat keys. (This is also the case for Werckmeister's tuning no.V, based on alternating 1/4-comma-tempered and pure fifths.)

Reflections

The typographical and other errors in the 1691 publication make it impossible to tell what Werckmeister's intentions really were, or how he came to find the monochord numbers that he wrote down, including the suspect D=176 that contradicts the principles of tuning he expounded. It is conceivable that he made an aural test via the monochord, mistaking the 175th division for the 176th, then carried through his calculations of tempered fifths and so on with this incorrect value — without checking carefully whether the resulting values were musically and mathematically consistent.

To test this possibility, one can examine the engraving of the monochord. The numbers used in the Septenarius are marked along the last line which is divided into 196 parts: the C at the far left is labeled 196, the C# is 186, and so on. But amazingly, the label '176' for D is positioned one place too far to the right — just where the number 175 ought to be! Count 'em:

Although this error seems somewhat incompetent, three things may help to excuse Werckmeister. First, he did not publish a table of major thirds — which would immediately have revealed the problem with D=176. Second, he might easily have believed he was hearing a wide tempered fifth at D-A even if he was playing a narrow one (with D=175) — it is an acoustic fact that simply by hearing the interval, one cannot tell a slightly wide fifth from a slightly narrow one. And third, the musical quality of the tuning with D=175 is fully consistent with Werckmeister's approving remarks, and appears equal or better compared to his other organ temperaments — which could scarcely happen by chance if its designer was incompetent, or obsessed with numerology at the expense of musical effect.

One question though remains – how easy is it to set this tuning on an instrument? Musicians are generally nonplussed by the mathematical presentation of the Septenarius, particularly the ratios which Werckmeister used to describe the tempering of fifths – far different from usual tuning methods that asked for intervals tempered by fractions (usually 1/4) of a comma. The tuning could have been transferred directly from the monochord to a harpsichord or spinet note by note (as could the other tunings of 1691). Although the monochord is never really accurate, since for example the string tension is slightly altered by the position of the moving bridge, this would have been an adequate first step, from which the beginner could learn how the various tempered intervals sounded. At this point the problematic value of D might become audible and good students might be able to correct it.

Setting the tuning on an organ would be trickier since it is very difficult to match an organ pipe to a plucked string (due to differences in harmonic content, dynamic and sustain) and conversely the correct degree of dissonance for tuning the tempered intervals by ear is hard, or impossible, to deduce from Werckmeister's ratios. The tuner would simply have to set narrowed, pure and widened fifths in the appropriate places, checking that the resulting chords round the 'circle of fifths' sounded reasonably good, and seeing at the end of the process (assuming that the tuning started with C) whether F and C made a pure fifth as they ought to. If not then part or all of the tuning must be readjusted.

If, though, anyone had attempted to use Werckmeister's original table of fifths to tune, they would be gravelled by the typographical mistakes in it, as well as the (apparently) incorrect tempering of D. Such mistakes make it extremely unlikely that any reader of Werckmeister's treatise did realise the tuning in any way on an organ. But they do not necessarily indicate that Werckmeister placed no musical value on it. The Septenarius is certainly no easier to use as a practical tuning than any of Werckmeister's other schemes from 1691: his claim that it was "very easy to set" appears as pure salesmanship. Having said that, it is no harder to produce a decent harpsichord or organ tuning using it than with his other schemes.

'Werckmeister III' seems to have attracted a lot of interest in the performance of late Baroque music; why not go a little further and try 'Werckmeister VI' for a change? It would have just as much musical justification. What does it sound like? Watch this space.