Why does the Venova have a couple of bad notes? This is a Yamaha instrument. They know how to build an instrument that is in tune (both with itself and to standard pitch). I theorized, then, that there was something in the physics of a woodwind that didn't permit all notes to be equally in tune.
Today I figured out how that works.
Start with the scale. We divide the octave, somewhat arbitrarily, into eleven equal parts. But the most-used scales we have built use only seven of those notes. There is a specific sequence of one-note intervals and two-note intervals that makes up the major scale, and by changing the starting note, all of the classic modes (the Phrygian and Lydian and so forth), including of course the minor scale.
The notes of a scale sound harmonious to each other; many melodies do not stray from the scale tones. Thus, the important tones for any instrument -- the ones you want to make easiest to play and most in tune -- are the scale tones.
In the case of a recorder, one finger and a thumb are used to support the instrument, one thumb covers the octave key; that leaves enough fingers for the scale tones but not enough for all the possible notes in an octave. So only enough tone holes are drilled to give you the major scale.
The out-of-scale notes, then, are achieved in other ways; in the recorder, by fork fingering and half-hole techniques. Or put another way, by manipulating combinations of the existing tone holes.
(Worth putting in here a bit about transposing instruments. Basically this idea of scale tones simple, chromatic tones harder, holds regardless of the instrument's range. A recorder in F plays the same scale sequence as the shorter recorder in C, just down a fifth. Some instruments take this one step further and are written in sheet music as if they are in C; like the Bb trumpet, which when playing a written C is actually playing Bb.)
So here's the issue. These scale tones have an imperfect relationship with the harmonic sequence. They are in fact (especially when you add in Equal Temperament) in rather arbitrary locations. The relationship of the chromatic tones has, unsurprisingly, no precise mathematical relationship to the tone holes drilled for those scale tone.
A wee bit on the acoustics of woodwinds. An oscillation started at the fipple, embouchure or reed is coupled to the column of air in the body of the instrument, creating a standing wave at some multiple of that original frequency. A set of harmonics, actually; by simple Fast Fourier transform, all of the harmonic series from the fundamental are represented (in different quantities which provide most of the distinctive timbre of that instrument).
When you open the tone holes sequentially from the bottom the acoustic length of the body changes, thus the frequency of the standing wave. But if you open mid-point holes in fork fingering, or half-cover a hole (both standard recorder techniques) you selectively enhance and suppress some of the harmonics, changing the flow pattern, causing the effective acoustic length to change by some fraction.
All of these special fingerings are, to repeat, built from the available tone holes; from tone holes that had been specifically drilled for the scale tones.
So it makes perfect sense that the fingering for most accidentals (another way to refer to out-of-scale tones) is a set of compromises. In the Venova, specifically, it can be set up for German fingering, with one set of notes generally flat, or for Baroque fingering, with an entirely different set of notes being off-pitch.
(Post written earlier and published today)
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