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In his Book Basics, Simon Fischer wrote in a footnote that:

The exact tuning of a sharp or a flat depends on the key, style and character of the music. For example, Bb as the tonic of Bb Major is higher than the Bb as the third of G minor. Tuning also depends on whether it is a single or a double stop, and what notes other instruments are playing

So my first question is, does this imply that accidentals need not be a particular pitch all the time? it is subject to other factors? There is no set frequency for an accidental unlike a normal tone, say for example: A=440? However, this matter is further complicated on another part in the same book; where it is written that:

The exact tuning of major, minor, augmented and diminished intervals depends, to a certain extent, on individual taste and the character of the music. For example, some players prefer wider (brighter) major thirds, or narrower (darker) minor thirds, than others. But perfect intervals- fourths, fifths and octaves- are either in tune or not, and cannot be adjusted according to taste.

So this time in stead of accidentals the focus is on intervals. In this case, if the piece starts with an accidental, i.e. Bb then there cannot be any variation for it. But this also implies that a natural note can be sharpened or flattened. In Bb Major, one might want to slightly sharpen the D note so that it resolves more easily into Eb, that is to say a wider major third.

So, my final question in relation to all of this is that, does this mean no note in music has a particular pitch/frequency that must absolutely be maintained? it is subject to the construction of the piece and other factors?

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There are a number of reasons why notes cannot be said to have exact pitches...

  • As well as just intonation which represents the purest, most consonant tuning possible, many different systems of temperament are possible. Equal temperament is particularly common these days, but in many situations musicians will still gravitate towards something closer to just intonation. This may mean that in a piece that modulates (or plays chords that don't come from one key only), the pitches of particular notes may shift slightly.

  • Musicians are not limited to the chromatic, diatonic, or any other scale; they are free to play any pitch they want for artistic reasons. This might be seen as microtonality in the case of the use of a fixed set of pitches outside the chromatic scale; however, some styles (blues being an obvious example) reject the idea of using only a fixed set of pitches, also using ranges of pitch in certain areas of the octave through which notes can be bent and inflected.

  • You mention A=440, but that's not fixed; there is a wide range of different concert pitches that are used.

Does this mean no note in music has a particular pitch/frequency that must absolutely be maintained? it is subject to the construction of the piece and other factors?

Yes. None of the reasons I've given are specific to accidentals.

  • A=440Hz may not be the industry standard, but it's probably the most prevalent. But - once any group of musos has tuned to a particular A, that becomes the datum point, at that time, surely? – Tim Apr 3 '16 at 8:42
  • @Tim I only mentioned it as the OP stated There is no set frequency for an accidental unlike a normal tone, say for example: A=440 - I thought he might mean he was seeing it as a strict absolute. Maybe I misinterpreted; I've relegated it to being my third point. In any case, in some keys the '440' A might itself be an accidental, rather than a 'normal note'. – topo Reinstate Monica Apr 3 '16 at 8:51
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There may be confusion between 'ordinary' notes (C, E, A -white keys on piano) and the # and b ( black keys).Don't think that 'ordinary' notes are any different from 'accidental' notes. When tuned to A=440Hz, then A will be that. Since, on violin, it's an open string, it can't be bent up or down from that, can it? I think what's trying to be said is that intervals rather than particular notes themselves can be (and are) stretched and squashed, particularly on unfretted stringed instruments, like violin. On fretted instruments it's still possible to move an interval slightly, but something such as a keyboard is set at 12EDO.Mentioning the Bb, it's its relativity to another note that's deemed important. In Bb, since it's the root, it will be pretty sacrosanct, whereas playing it as a minor third in G minor may call, to the player's discretion, for it not to be exactly the same pitch as the former Bb.

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I would answer your final question with yes: there are no absolute pitches or frequencies that must be maintained for particular notes, unless you choose to do so. How to decide what frequency a note should have depends upon taste, style, and practicality. And it's not a simple thing to decide.

Nowadays, the most "official" frequencies of pitches in Western music are considered to be those dictated by 12 tone equal temperament- that is, the semitones are separated by a factor of the twelfth root of two. Most modern keyboard instruments are tuned this way.

https://en.wikipedia.org/wiki/Equal_temperament#Twelve-tone_equal_temperament

But as your quotes point out, this is not the whole story, especially for instruments where intonation is flexible- musicians tend to bend pitches to fit their tastes and perceptions. I actually bend my major and minor thirds in the opposite direction that Fischer suggests: he is obviously going for more contrast between major and minor thirds, not more consonance. But this is a matter of taste.

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