Tuning, temperaments and the traverso (1)
(this consists of three pages 1 2 3)
Introduction.
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.
The Kirnberger III is a great example of how a temperament can be constructed. For detailed instructions on how to tune this temperament on a harpsichord, please refer to [8]. The basic idea behind Kirnberger III is as follows:
In the circle of fifths, begin with a pure major third on C, which gives you an E two octaves higher. Then, construct the next eight fifths, adjusting only one of them to return to C.
The first step involves calculating the frequency multiplication factor needed to reach E two octaves above C:
2 × 2 × (5/4) = 5
This value—5—is the total multiplication factor, and we’ll call it M1. The 2 × 2 accounts for the two octaves, and 5/4 represents the pure major third.
Ideally, we would like the eight fifths to be pure. A pure fifth has a ratio of 3/2, so eight pure fifths would result in:
(3/2)8 = 25.6289
Combining this with the third (M1), the total becomes:
(3/2)8 × M1 = 25.629 × 5 = 128.14544
However, to complete the circle of fifths correctly, we should land exactly at:
27 = 128
This slight discrepancy means we must adjust one of the fifths. In the Kirnberger III temperament, the solution is to make the fifth between F♯ and D♭ slightly smaller, while keeping the rest pure.
Next, we define two more multiplication factors:
- M2: the factor to go from E to F♯, which is (3/2)2
- M3: the factor to go from D♭ back to C, which is (3/2)5
To complete the cycle, we require:
M1 × M2 × X × M3 = 128
Where X is the unknown multiplication factor between F♯ and D♭. Solving for X:
X = 128 / (M1 × M2 × M3) = 1.4983
This value of X represents the adjusted fifth in Kirnberger III, and it turns out to be very close to an equal-tempered fifth:
1281/12 = 1.4983
Remarkably, the result aligns almost exactly with equal temperament—quite fascinating!
Lastly, we need to consider the fifths between C and E. Since M1 corresponds to four such fifths, they are made equal, each with a multiplication factor of:
M11/4 = 1.495
This is slightly smaller than the equal-tempered fifth.
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.
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(this consists of three pages 1 2 3)
