The Importance of Octave Stretch
What is perfect flute intonation? One might reasonably say that if an electronic tuner says every note is spot on, without needing special attention from the player, that flute has perfect intonation. However, in measuring flute tuning it is often the case that the second octave is a bit sharp relative to the first. This is referred to as a stretched octave, and there is a good reason for it.
A classic flute exercise is to play a long note starting in the first octave, then gradually increase the volume and pressure until the note switches to the second octave. Ideally, this will be a smooth transition with good tone quality throughout. As flutes tend to play sharper with increasing breath pressure, typically the pitch of the lower octave note will slowly increase up to the point where it transitions.
A flute with a flat second octave makes this exercise somewhat frustrating. As pressure is increased, air noise increases, and out-of-tune harmonics will beat against each other. So it's not just a matter of upper octave tuning; the tone and response of the lower octave note suffers as well. Empirically, the cleanest transition occurs when the second octave is just a bit sharp. There is an excellent discussion of this in Chapter 22 Section E of Fundamentals of Musical Acoustics
[Benade2]
. There is also an interesting discussion on Chiff+Fipple
[CF2]
that touches on several of the topics on this page.
In addition to the physics of the response, there is also the psycho-physics. Physicist and musician Juan Roederer has written a great book on how humans perceive musical notes. In a group of players, if one is playing a bit sharper they tend to sound better. Such factors may play a role in instrument design.
It seems to me that a good octave stretch is particularly important for the lowest notes of the flute. A longer vibrating air-column responds more slowly and weakly than a short one (for a given diameter). This has to do with the aspect ratio of the vibrating part of the bore (the ratio of length to diameter). A larger octave stretch may be desirable for the lowest notes, to strengthen their response. Consider the low D in Irish flutes. The Irish style of playing traditional tunes includes accenting the low D in particular with a lot of breath pressure. This produces a "buzzy" note that is a mixture of the first and second octave note. In order for this to sound clearly--given the extra volume--a wider octave stretch would seem desirable.
Although the relationship between the first and second octave gets the most attention, the same principle applies to the third octave. It's important to note that the third overtone is part of the first octave note's overtone series (1, 2, 3...) but not part of the second octave series (2, 4, 6...). Because of this, I think it makes sense to calculate the third octave stretch relative to the first octave rather than the second. Also, since musical tones contain higher harmonics of the fundamental, the fourth and higher harmonic resonances have an effect on note quality as well. The ideal state is Harmonic Alignment (ommmm.....) where the harmonics of the blown note correspond to the body resonances of the flute.
Speaking of the third octave, it should be noted that the classical conical bodied flutes were intended for classical music, and that carefully crafted compromises were aimed at improving the tuning of particular third octave notes. Modern use of this type of instrument for folk music doesn't use the third octave much. This opens the way for a re-design that eliminates those compromises--something of a devo-lution!
A good example of this is cork position. Cork position primarily affects notes B5 and above. In Irish flutes it has been found that pulling the cork out until B5 just starts to go flat has a good effect on low register response. This has the effect of making C#5 rather flat, but in that genre of music this is considered a worthwhile tradeoff.
So, in summary, there is some ambiguity about what constitutes the (or should we say "a") best tuning, and compromises are generally necessary. The style in which a flute is played may influence how its response should be tailored. I think this is relevant in optimization, where it is necessary to explicitly specify the desired tuning relationships (or else risk having them varied arbitrarily).
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