A sensitive note is a note that is to be played somewhat sharper than its normal or written pitch.
The English flutist R. S. Rockstro was a strong advocate of the Boehm flute and equal temperament, but he would still write in his A Treatise on the Flute (1890):
...there are certain cases in which a slight and temporary departure from [equal temperament] may be permitted. These cases result in the lessening of a semitone, either by raising the lower, or depressing the upper, of the two notes which form the boundaries of the interval ... This reduction of the semitone, if not carried to an extreme, is generally as agreeable to the ear as the slightest enlargement is offensive, but on no account must a note be raised or lowered when it forms an integral part of the accompanying harmony, or when it is near to a similar note which not so altered from its normal pitch. The raised lower note as the semitone is known in France, where it is much used, as the "note sensible," or sensitive note. I know of no technical name for the depressed upper note of the interval.
Here is one of his examples. Rockstro uses "S" to indicate a note to be sharpened and "F" a note to be flattened.
In Henri Altès' Méthode de flûte (1880, English translation 1906) for the Boehm flute, we read:
"THIS NE PLUS ULTRA of intonation [the sensitive note] is easily obtained by the voice as well as by stringed instruments ... which gives them a great superiority over instruments with fixed sounds, nevertheless; upon certain wind instruments [presumably including the flute] they have succeeded in satisfying the most delicate ear upon this point by means of irregular fingerings."
Here is link to Altès' table of sensitive fingerings for the Boehm flute, some of which use take advantage of the perforations in the keys of French model Boehm flutes. It is necessary to provide "irregular" fingerings for the sensitive notes because the embouchure alone cannot manage to make these required pitch changes fast enough. In addition, the sensitive fingerings are often chosen to have a softer tone.
While the above-quoted excerpts are from late 19C sources, sensitive notes (on simple system flutes) were in extensive use early in the 19th century. Berbiguier's Méthode pour la flûte (1818) devotes a full 25 pages (out of 260) to fingerings and illustrations of sensitive notes in context. Though he does not use that term, Nicholson's Preceptive Lessons (1821) include many fingerings for notes intended as sensitive notes, as his examples of their usage make clear. Here is a good illustration of the use of sensitive notes in a short passage from a flute duet by Walckiers. (The "S"s are my additions; printed music never indicates where sensitive notes are to be used.) See the footnotes for a link to an extract from Walckiers.
The fingerings just above are among those that occur in Walckiers' Méthode de Flûte. Op.30 (1829). Here is a link to a facsimile of his 11 page chapter on "Des Demi-tones Alteres".
The term "sensitive note" literally means "leading tone", the seventh degree of a major scale. Many musicians find it expressive to play leading notes somewhat high in melodic passages. Raised notes (and other expressive fingering) were also found useful in many other situations in the 19th century.The terms "sensitive note" and "notes sensible" are common in English and French sources, so I use that term here and in the associated chart even though I would prefer the term "augmented note" (as used by T. Lindsay in his Elements of Flute-Playing, 1828–30). I have not seen references to the lowered notes that Rockstro mentions, but I have seen them used in 19th century exercises and would suggest the name "depressed notes". The sensitive note is always the lower note of a semitone where the notes have different letter names, like from C♯ to D or B♭ to A or E to F, but never C to C#. It is only natural and sharped notes that have sensitive fingerings. Sensitive notes are used mostly in the second and third octaves.
The pitch difference between a sentitive fingering and a more standard one varies tremendously from note to note and flute to flute, but is often about 1/3 of a semitone and sometimes more. Almost always, the transition between the augmented note and the note written as the semitone above involves easy finger movements, and the sound transition is smooth, without crossing any register breaks when possible. In addition, there may be a softer and distinctive timbre for the augmented note, which is desirable and adds to the effect.
I should mention that many flute methods do not mention sensitive notes at all.