More on Handel's Temperament
15 September 2019
(click on the headings below to access the desired section)
Introduction and background
Figure 1. Tuning instructions for Handel's Temperament
In 2016 I posted an article on this site about the keyboard temperament which is popularly associated with Handel
. Although the connection with the great man is tenuous at best, the temperament is popularly referred to as
'Handel's' and it is denoted thus here. It was published around 1780
 and it is interesting in that detailed tuning instructions were given (Figure 1). However these strike us today as imprecise because they lack quantitative information such as beat rates, instead relying on vague phrases such as 'tune the fifth pretty flat'. Consequently it has hitherto been impossible to recover a single, definitive version of the temperament from the instructions. This has led various authors to hazard diverse guesses as to what the temperament might be, ranging from sixth-comma meantone at one extreme to equal temperament at the other.
This section outlines the approach which was developed to assess how well a given temperament matches the tuning instructions of 1780. A method was sought which would enable various temperaments to be ranked objectively in terms of how well they corresponded with the instructions, and this implied the use of some sort of numerical marking scheme. Given the woolly nature of the instructions it was not immediately obvious how this could be achieved. However a solution emerged by observing that the verbose instructions could be reduced to a set of 6 simple descriptions of interval quality. The tuning instructions associate these with intervals within the circle of 12 fifths which define any temperament. By 'interval quality' is meant the amount by which each fifth deviates from pure, a condition in which no beats are heard. The deviations can only be assessed relatively rather than absolutely in view of the absence of quantitative content in the instructions which do not mention beats at all, let alone how fast or slow they should be. In other words, it was only possible to assess the beat rates of each fifth relative to the others ('faster', 'slower' or 'the same') rather than specifying each beat rate numerically as we do today. However it turned out that this was an unexpectedly sensitive means of ranking candidate temperaments against each other since the relative beat rates of the intervals in the circle of fifths were quite critical if all 6 tuning criteria were to be satisfied simultaneously. This explains why tuning methods which appear hopelessly vague at first sight, such as those here, held sway for centuries before accurate beat timing progressively entered tuning practice in the nineteenth century. Although each tuning instruction in Figure 1 is indeed vague in isolation, the full set enables a reasonably accurate and repeatable outcome to be achieved by an experienced tuner when they are taken together.
In addition it is stated that 'all fifths should be flat', and therefore none should be pure or sharp as in some other temperaments. Jorgensen
 also identified a similar set of 6 criteria whereas Johnson
 reduced her list to 5. I used those listed above since they drop out from the tuning instructions immediately without needing further explanation. These descriptions of interval quality therefore identify a number of apparently differently-tuned fifths based on what a tuner hears in terms of their relative beat frequencies. It is important to note that they do not immediately relate to interval sizes expressed as ratios, usually in cents. If two audio frequencies are denoted by
f1 and f2, one hears the difference between them (f1 - f2) as a beat when tuning by ear, not their
(f1/f2). Beats are a direct and audible measure of frequency differences whereas frequency ratios cannot be determined aurally, notwithstanding their usefulness when studying temperaments from an arithmetical viewpoint. I labour this simple but vital point because it is often overlooked, and I shall return to it presently.
Table 1. Correspondences between the fifths and the descriptions of interval quality in Handel's temperament
The next step in assessing how well a given temperament corresponds to the tuning instructions is to match the numerical beat frequencies of its fifths against the descriptions of interval quality. This will now be done for a number of temperaments and the results will then be compared to see which one wins the competition.
Johnson  was one of those who maintained that Handel's temperament is actually sixth-comma meantone in disguise, a grossly unequal temperament used by Gottfried Silbermann in the eighteenth century. It is easy to disprove this using the method just described. Table 2 shows the correspondences between the beat frequencies of sixth Pythagorean comma meantone temperament, the intervals comprising the circle of fifths and the descriptions of interval quality in Handel's temperament. Here the intervals are confined to a single octave by inverting alternate fifths into fourths. This can be done since the note-names remain the same, and both a fifth and its inversion beat at the same rate.
Table 2. Sixth-comma beats versus descriptions of interval quality in Handel's temperament
Note firstly the Wolf interval between
E♭ and G#. Unlike all the others this must be tuned (grossly) sharp rather than flat, thereby conflicting with the instruction which requires all fifths to be flat or narrow from pure. The remaining intervals have beat frequencies which do not accord at all with the descriptions in the instructions. For instance, C-G should be tuned 'pretty flat' yet it beats slowest in the entire set, indicating that it is closest to pure. On the other hand B-F# has the fastest beat yet the instructions stipulate that it should only be 'nearly as flat' as C-G. Other contradictory examples are the intervals in the three lowest rows of the table which should have 'a little flatness', whereas in reality they are all significantly flatter than the 'pretty flat' C-G! Against this evidence it is therefore difficult to conclude that the tuning instructions relate to sixth-comma
It has been suggested that Handel's temperament was an attempt to move closer to equal tuning at a time when composers were seeking the freedom to modulate more freely between keys than the old meantone temperaments permitted. One piece of evidence supporting this is the statement in the Handel instructions that 'all thirds must be tuned sharp more or less as all fifths should be flat'. This certainly applies to equal temperament, though also to others. However elsewhere the instructions say that 'the fifth will not bear to be reduced so much below its true accord as the third will to be raised above it'. Presumably this means that the ear will not tolerate the beat frequencies of the fifths approaching those of the thirds, in other words the fifths must not be narrowed too much from pure whereas (in the words of the tuning instructions) the thirds may be allowed to become 'considerably too sharp'. Equal temperament also satisfies this constraint. Unfortunately these qualitative observations still remain some way from confirming that Handel's temperament is equal, but by following the same method applied above it is possible to decide the matter conclusively. Table 3 shows the correspondences between the beat frequencies of equal temperament, the intervals comprising the circle of fifths and the descriptions of interval quality in Handel's temperament.
Table 3. Equal temperament beats versus descriptions of interval quality in Handel's temperament
Equal temperament beat frequencies are exactly half those of sixth Pythagorean comma meantone, though note that the values in these tables are rounded to only one decimal place. (Nor does equal temperament have a fast-beating sharp Wolf). Consequently it is unsurprising that the scores in the right hand column, which largely depend on relative beat frequencies, are much the same as for the sixth-comma case (Table 2). The exception is that D-A, beating once per second, can reasonably be regarded as 'a good fifth' and it has therefore been assigned a tick rather than a cross. Nevertheless, equal temperament also scores dismally overall with a mark of 1 out of 9.
Handel's temperament was studied in detail by Jorgensen  who concluded that it was mildly unequal, though his approach was flawed and unscholarly. Ignoring the lack of evidence he insisted that the temperament was indeed by Handel, who (according to Jorgensen) tuned entirely melodically without testing intervals! He declared baldly that two of the fifths 'were no doubt just' without explanation. He devoted much effort to specifying 'six different sizes of fifths' in terms of frequency ratios expressed as fractions of the Pythagorean comma. Thus he fell into the trap mentioned earlier - he overlooked that tuners can only hear the difference between two frequencies as a beat when tuning by ear, not frequency ratios. The descriptions of interval quality in the Handel tuning instructions are plainly aimed at practical tuners who can only attempt to rank the fifths based on the beats they hear, regardless of whether they try to count and time them. Therefore the ranking is not based on their sizes measured in terms of commas which, in the eighteenth century, would have mainly interested numerologists rather than tuners . Although Jorgensen did describe a tuning method using beats, he only derived the beat frequencies mathematically after having first invented an artificial interval ranking based on cents rather than simply ranking the beat frequencies in the first place. Given this convoluted path, it was therefore unsurprising that the result was not a particularly close match to the interval descriptions in Handel's temperament, as can be seen from Table 4. This shows the correspondences between the beat frequencies of his temperament, the intervals comprising the circle of fifths and the descriptions of interval quality in Handel's tuning.
Table 4. Beats in Jorgensen's temperament versus descriptions of interval quality in Handel's tuning
Notwithstanding the foregoing criticisms, with a score of 6 out of 9 it achieves a much better rank than the sixth-comma and equal temperaments discussed above. However, and fortunately for Jorgensen, his questionable inclusion of the two pure intervals (C#-G# and E♭-G#) does not pull his score lower because they are not addressed explicitly by the tuning instructions. The major remaining downside is that the 'pretty flat' interval (C-G) is not flat enough. This results in its relativities with D-G, F#-B and C#-F# (all of which should be less flat) also being degraded. Had Jorgensen concentrated less on cents and more on beats he could have arrived at a better solution. Nevertheless, having tried the temperament I find it pleasing despite Jorgensen's odd approach. However his data seem to be used in most if not all digital tuning aids which claim to tune Handel's temperament, and in view of this it must be repeated that they do not result in a particularly close match to the tuning instructions in terms of relative beat frequencies.
This has been mentioned already. It was my first attempt at realising a version of Handel's temperament and full details are in another article on this website . They need not be repeated here so we can cut to the chase and tabulate its characteristics in the same way as before. Table 5 shows the correspondences between the beat frequencies of the temperament, the intervals comprising the circle of fifths and the descriptions of interval quality in Handel's tuning.
Table 5. Beats in 'Pykett 2016' versus descriptions of interval quality in Handel's tuning
Like Jorgensen's, it is a similarly pleasing mildly unequal temperament and it achieves the same score (6 out of 9). Unfortunately it also shares the same problem in that the 'pretty flat' interval (C-G) is not flat enough, thus perturbing the relativities of several other intervals which are compared with it in the tuning instructions. This persuaded me to develop the later version described below.
This is the temperament outlined in the
Organists' Review article  and it is now described in more detail. Since arriving at the earlier version in 2016 I have studied the attempts of other authors together with several other candidate temperaments in use around 1750. Among other things I concluded that temperaments could only be assessed against the Handel tuning instructions in terms of their relative beat frequencies, which is the approach applied several times already in this article. It also became clear that the main problem in achieving an optimum match to the tuning instructions of 1780 hinged on the issue of the meaning of 'pretty flat' applied to the interval C-G. It is an important interval in Handel's temperament because the temperings of three others are compared with it in the tuning instructions. So if we don't get C-G right then we won't get the others right either. In theory one could flatten it by an arbitrary amount, but in practice this is not possible. If all fifths are to be flat as the tuning instructions insist, there is little room for
manoeuvre. In all temperaments the deviations from pure for all 12 fifths must always add up to the Pythagorean comma (-23.46 cents). Consequently one cannot compensate in this case for excessively flattened fifths by inserting pure or sharpened ones elsewhere around the circle. Otherwise it is all too easy to end up with a sharp Wolf interval or several pure ones, as Jorgensen found. Not only does this run counter to the 'all fifths should be flat' constraint, but it has been shown above that strongly unequal temperaments with a sharp Wolf do not fit the tuning instructions at all (see Table 2 relating to sixth-comma meantone for an example).
Table 6. Circle of fifths tuning data for 'Pykett 2019'
This version of Handel's temperament (called 'Pykett 2019' for convenience) was developed by assigning simple fractions of the Pythagorean comma to each tempered interval around the circle of fifths as shown in the third column of the table. As mentioned above, the interval C-G was flattened by nearly one quarter (in fact 0.24) of the syntonic comma (-21.51 cents). This is virtually the same as that of the tempered fifths in quarter-comma meantone tuning. Having set this as a datum value, the remaining fractional values were adjusted until agreement was obtained between the relative beat frequencies of the intervals and the qualitative descriptions of interval quality in the tuning instructions of 1780. Table 7 shows the correspondences between the beat frequencies of the temperament, the intervals comprising the circle of fifths and the descriptions of interval quality. Note that this approach was fundamentally different to that of Jorgensen. He completed his circle of fifths first by ranking all the intervals in terms of ratios before translating them into the corresponding beat frequencies. Here I only adopted a single datum value for the flattest fifth (C-G) expressed as a frequency ratio. The remaining ratios were then calculated after having first adjusted the beat frequencies to match the tuning instructions. Put simply, Jorgensen developed his temperament on the basis of ratios whereas mine was guided by beats from the outset.
Table 7. Beats in 'Pykett 2019' versus descriptions of interval quality in Handel's tuning
The outcome was yet another pleasing mildly unequal temperament but unlike the others described earlier it now scores with top marks - 9 ticks out of 9 in the right hand column of the table. That the temperament could be set up successfully from a single quarter-comma datum suggests that it might have evolved from quarter-comma mean tone tuning.
If so, the evolution process was probably empirical in which the interval C-G was left as the sole quarter-comma fifth while the others, save the Wolf, were widened closer to pure. This process would also have resulted in the Wolf itself
(G#-E♭) disappearing as a grossly-sharpened fifth and re-emerging as a slightly flattened one so that the temperings of all twelve fifths would continue to add up to the Pythagorean comma as required. Table 6 shows that the former Wolf is now one of several equally-tempered fifths in this temperament, all narrowed or flattened by 1.955 cents. Adjusting the temperament was found to be quite critical if all of the interval quality descriptions in the original tuning instructions were to be satisfied. Therefore, although the instructions are vague, they leave less room for manoeuvre than might appear at first sight. As mentioned previously, this explains why an imprecise set of tuning instructions can nevertheless lead to a useful outcome in the hands of an experienced tuner.
Table 8. Chromatic note data for 'Pykett 2019'
Alternatively, for those who dislike tuning digitally the temperament can be set up by ear using the beat frequency data in Table 7. A principal-toned stop should be used if tuning an organ. The intervals in the table are shown as a tuning sequence in Figure 2 below. As in the table the intervals are confined to a single octave by inverting alternate fifths into fourths. This can be done since the note-names remain the same, and both a fifth and its inversion beat at the same rate.
Figure 2. Tuning an octave using fifths and fourths
A numerical marking method for ranking various candidates against the tuning instructions for Handel's temperament has been described. Several temperaments have been examined in detail and the marks awarded are summarised in Table 9 below:
A wide spectrum of different temperaments was represented in the competition, so the fact that the three mildly unequal candidates were the only ones to achieve a respectable score is unlikely to be down entirely to chance. It therefore seems reasonable to suggest that the designer of Handel's temperament, whoever they might have been, was aiming for a 'well' temperament. However it is unlikely that they had equal temperament in mind because there are some features of the tuning instructions which clearly work against it, such as the insistence that the interval C-G should be 'pretty flat'. Any competent tuner in the eighteenth century would have known that equal temperament has no such intervals. Therefore the outcome can only be a mildly unequal temperament with a noticeable smidgeon of key flavour to add interest, and this aspect will now be discussed.
Key flavour is an elusive and largely subjective matter though it is influenced by some objective factors as well. One is that the effect of temperaments set up on the organ depends on the player's choice of stops, and in this respect it is unique among keyboard instruments. This phenomenon cannot be treated here but a detailed article is available elsewhere on this site . All keys in Pykett 2019 are entirely useable and therefore it is a 'well' temperament with a pleasing range of subtle key flavours. In mildly unequal temperaments such as this one, the spectrum of flavours arises more from variations in the beat rates of the intervals in different keys rather than from the absolute tuning of individual notes deviating noticeably from expectations. Table 7 shows the beat frequencies for the fifths and fourths in the octave beginning at middle C for an 8 foot stop tuned to A440. The slowest beat (one every two seconds) is between E flat and B flat, indicating that this interval is almost pure, therefore it accords with the instruction that it should have 'a little flatness'. No beat in equal temperament in this octave is as slow as this. Several others are almost as slow. This factor might be responsible for the sort of 'gentleness' in the sounds of certain keys which I have noticed. The fastest beat (well over two beats per second) lies between C and G, thereby corresponding with the instruction which says it should be 'pretty flat'. It is faster than any in equal temperament (see Table 3), verging on the intrusive to my ears especially as it gets faster still in higher octaves and pitches. It is flattened by nearly a quarter of the Pythagorean comma whereas all the fifths in equal temperament are only flattened by one twelfth. However the upside is that the major third on middle C (C-E) now beats relatively gently at 8 Hz, slower than any others both in this and equal temperaments. These two intervals (C-G and C-E) account for part of the key flavour spectrum of the temperament. As to the rest of the major thirds, their beat rates jump up and down as one moves across the octave, whereas in equal temperament they vary gradually and predictably. Not surprisingly, this contributes to a more interesting aural experience compared with the blandness of equal tuning. This is shown graphically in Figure 3 below.
Figure 3. Beat frequencies of major thirds and fifths for equal temperament (left) and 'Pykett 2019' (right)
Here the beat frequencies of the fifths and major thirds are plotted against each semitone of the chromatic octave starting at middle C (again for an 8 foot stop tuned to A440). The left hand curves relate to equal temperament and the right hand ones to Pykett 2019. The mathematical perfection of equal temperament is reflected in the smoothness of its curves, whereas for Pykett 2019 this has been completely disrupted. Consequently the ear is not able to predict how the beat patterns change abruptly from key to key, unlike in equal temperament where the deterministic progression across the octave leads to a more anodyne listening experience. As described above, the temperament was developed largely by trial and error by matching beat frequencies to the set of hazy tuning instructions. No attempt was made to
'force' a theoretically attractive result in a mathematical sense, an aspect where I feel Jorgensen
 led himself astray in developing what he called 'the theoretically correct' approach.
With such a foggy set of tuning instructions, there cannot possibly be anything
like a theoretically correct result - the mere phrase is meaningless. Owing to the clear distinction between the two sets of curves, the ear has little difficulty in detecting the differences between Pykett 2019 and equal temperament.
The main outcomes of this work
1. "Handel's temperament
revisited", an article on this website, C E Pykett, 2016.