Tuning, temperaments and the traverso (1)

(this consists of three pages 1 2 3)


In the eighteenth century there is a gradual change in the appreciation of the one-keyed flute. In the beginning it is the new sweetly voiced instrument. It is pushing the recorder completely out of the picture. However during the eighteenth century there is more and more criticism. This criticism aims both at the vagueness of certain tones and at the tuning problems. Mozart shows his appreciation in a well-known citation (i.e. [1]) for the extraordinary fact that a certain Wendling was able to play in tune. The flute amateur Ribock writes in his book [2] ‘leaving small imperfections in the intervals aside I only need to mention the f/1 g♯/1 and b♭/1. These three tones are so extremely miserable that I wonder why nobody had the idea before to improve them by adding a special mechanism, such that they can be made higher or lower in a natural way. ‘ This is a citation from the end of the eighteenth century, a time where playing equally easily in all keys became a requirement. Therefore, and for other reasons (see history) more and more keys were introduced. In our time, 1938 Adam Carse [3] writes ‘to the collector they have an indefinable charm; to the modern player they are useless old things which, if they aroused any interest at all, awaken only feelings of sympathy for the old players who had to manage with such inferior instruments.'
Fortunately, at present we have a different opinion on the traverso. The above remarks should be placed in the light of the prevalent opinions at the time they were made.
We should be aware that something has changed in our view of the passed. Until recently periods of the past were interesting from time to time but mostly to be improved upon. Now we are trying to reproduce eighteenth century music in, whatever that means, the original way. Improving is seen as "probably not as good as original".
A second issue is the two ways of viewing the qualities of a traverso. In one view the difference in tone quality is merely a deficiency, in the other view it is also typical for the charm of the instrument.
For many, the deficiencies do not prevent a preference of the traverso for eighteenth century music above the modern flute.
To understand the criticism better I will consider tuning the traverso and some of the typical temperament properties more closely. Tuning, of course for me as a flute maker is the basis. However when tuning I am confronted with the choices to be made, and therefore with the problem of temperament. Therefore, first a few general remarks about temperament. I am not a musicologist and therefore do not pretend to present a scientific account here.
Equal temperament revisited.
In our world we are used to the temperament of the piano as if this represented a law of nature. First of all we should understand that "in tune" as an absolute concept does not exist. This can be understood by considering octaves and fifths. A pure octave doubles the frequency of a tone. A pure fifth multiplies the frequency of a tone with 1.5. It is easy to see that 12 fifths should exactly produce 7 octaves. Alas with pure fifths this is not the case because (1.5) 12 =129.7463..  and  27 =128. I know that the non-mathematically inclined hate this kind of calculation but maybe the following will make it more understandable. Each time we go a fifth higher the frequency is again multiplied by 1.5, hence for twelve fifths the (1.5) 12.
Each time we go an octave higher the frequency is multiplied by 2, therefore seven octaves produce 27. In any case pure fifths and pure octaves do not go together.  Similarly for pure thirds ((5/4)3 = 125/64 which should be 128/64). In short a pure tuning does not exist. This has been known for many centuries. Therefore there have been many compromises. The temperament fanatics have calculated those temperaments in the past in many decimals that nobody ever can hear. In these temperaments some intervals usually are made purer at the cost of others. For instance in mean tone temperaments certain thirds are chosen to be pure at the cost of certain fifths being terrible. 
Equal temperament is one such compromise. It is a compromise where the twelve tones are all equally out of tune in the following sense. For a half tone distance,  to get the highest the same number always multiplies the frequency of the lowest note. Twelve half notes must give an octave. The one thing that is kept pure in all these temperaments is the octave. Therefore the above "same number" twelve times multiplied by itself must give 2. This means that in equal temperament a the frequency of a  tone is multiplied by 21/12 to go a half tone higher.                          
 Another way to see this compromise is the following.  All fifths are made an equal amount slightly smaller such that indeed 12 fifths give 7 octaves. So the frequency is no longer multiplied by 1.5 for a fifth but by 1281/12 =1.49831....
In the eighteenth century equal temperament was known but not really applied. One reason was that tuning equal temperament was not mastered yet. An extensive discussion of this issue can be found in [4], together with much further information about temperaments. Because temperament is a question of compromise taste plays a vital role. This is illustrated by the following. In the beginning of the eighteenth century a sharp was played lower than the flat. At the beginning of the nineteenth century sharps were played higher flats.
Temperaments revisited.
Let us distinguish now between the instruments with twelve fixed notes such as the harpsichord and the instruments with continuous tone possibilities such as the flute and the violin and of course the voice. The latter adapt all the time to the harmonics involved by using many different tones, the first cannot. So for the first fixed compromises for the thirds, fifths etc. are needed. Such a compromise is called a temperament.
Just a remark on the way we can adapt the intonation when playing a traverso. Of course the extent to which this is possible is limited. Traverso therefore also have a fixed temperament to certain extent. I always play as much as possible without compensating when I start tuning a new instrument. This is easier said then done because after twenty minutes I am adapting and compensating whatever. I therefore tune always over more days, relatively brief periods. In the end there will be notes that nevertheless need some compensation. Then comes what I find an extremely important property of a traverso. How easy is it to change the frequency of a note with the usual techniques. For instance if f# is a bit low is it sort of stuck there or is it very easy to play it higher. A traverso should be very flexible in this sense!

I have not seen a study dealing with the continuous way flutes etc. adapt and would be interested to find one if it exists. In the end, however this is the key question because with the embouchure we are able to play notes up and down very considerably. It seems to me that flute players playing together with a keyboard instrument adapt to the bass line. They will probably try to play pure thirds with respect to this bass line. Therefore they will sacrifice melodic pure intervals to harmonic pure intervals. This of course may clash with the temperament of the keyboard instrument itself. How this is resolved I do not understand, except for the fact that it seems that the keyboard player sometimes leaves the thirds to the flute player to be played pure rather than in the keyboard temperament. What the flute player chooses to do when there is no bass I would like to understand better but at present it is not clear to me.

To understand temperaments it is usual and convenient to consider them from the so called circle of fifths. Twelve fifths (should) build seven octaves and because they have not found an octave of the starting note with less than twelve they necessarily must have passed all twelve notes. So by considering twelve fifths we have a circle with all twelve tones we  have on the usual  fixed tone instrument. Therefore any temperament can be described by describing the size of the twelve fifths involved. There are many other ways, but let us stick to this.
The fifths are on e♭,b♭,f,c,g,d,a,e,b,f♯,c♯,g♯.
The twelve fifths overshoot the seven octaves as shown above. The difference is called the ditonic comma.  Distributing the ditonic comma evenly over twelve fifths gave equal temperament. Let us now consider another important inconsistency. Four fifths (should) give two octaves plus one major third, again for instruments with twelve fixed tones.  However, they do not. Again there is an overshoot (easily calculated as done above). This is called the syntonic comma. We may distribute this syntonic comma over the four fifths. However this can only be done twice with four fifths, because the third time four we have to close to come to an octave again.  If we take the eight fifths with a quarter syntonic comma to be consecutive this gives eight  pure major thirds. Pretty good. However the last four now have to bridge a pretty nasty gap to come back to an octave after twelve fifths. It is the play with those four last fifths that make the meantone temperaments a whole family, contrary to for instance equal temperament because there we have fixed all twelve fifths.  If we take three of those with again a quarter comma it is  Aaron's quarter comma or meantone temperament. However whereas the syntonic comma  is less then 1/4th of a half tone (22 cents) so each fifth has less than 1/16th (5.5 cents) of a half tone now we have for the last, twelfth  fifth a really big gap more like a quarter tone (36.5cents).
Now comes a crucial aspect in all temperaments that I have not mentioned so far. If we agree on a tempering of fifths according to a certain pattern, it still is to be decided where the pattern starts in the circle of fifths. Because they are all equal this is not important for equal temperament, but in all others they are different. It is often convenient to start the pattern at E♭ or B♭ because that way the bad intervals are less used. However I guess that the  bad thirds and fifths were every now and than used on purpose.
It will be clear that variations on this theme to improve the bad intervals thereby deteriorating the perfect ones can be infinite. This is the whole essence of temperaments: as soon as a number of intervals are perfect others are so bad that they have to be improved a bit etcetera.

We have to consider now the use of a temperament for making music! In the end that is what it is all about. Otherwise we could just as well say that equal temperament is the best because imperfection on each fifth is minimised in the maximum deviation sense. However the character of different keys was used to express certain feelings, flavours, emotions and what one does with music. It should be realised that the fact that pure intervals are good is subject to acquired taste! This is demonstrated by the fact that many equal tempered piano players are not charmed by pure thirds at all! So the not so pure intervals were used on purpose. Different keys making different music is lost in equal temperament (except for major and minor of course). When searching again for the most appealing expression of old music by playing on copies of old instruments one clearly has to try out the use of old temperaments as well. And equal temperament is not the best because it does not bring out the good and it may not even show the bad enough.
One more temperament should be considered briefly in this context because it is the one we will finish with as close to the traverso. It is what is usually called the Rameau 1726 temperament (complete description with beats for tuning) also called "temperament ordinaire" . This temperament is described by Rameau in a way that allows more interpretations. However I am taking one that is derived from his descriptions by the real experts in this field (which I am not) and as said indicated by this name. It starts on B♭ with seven consecutive fifths diminished with a quarter syntonic comma (b♭-f-c-g-d-a-e-b), the others ( b-f♯-c♯-g♯) pure and (g♯-e♭-b♭) equally wide to fill up to the octave. This gives four pure thirds where it counts most , on b♭,f,c,g and an almost pure third on d. 
We will see more of it in the following. The difference in cents deviation wit respect to equal temperament between the graph below at the end and the values in the above link is only due to the 'pitch anchor', so to speak.  The graph has a for deviation 0, the link has c for deviation 0.
Basic concepts in traverso tuning.
(this consists of three pages 1 2 3)

Simon Polak: Early Flutes

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