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I've heard it said that, whilst on most instruments these notes are played with the same fingerings/technique/etc there is a subtle difference.

This isn't specific to this particular note combination, but to all enharmonic equivalents.

What might this teacher have been referring to?

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It may be that the only way for me to find out is ask the person in question what they meant... –  8128 Apr 26 '11 at 19:04
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I had a guitar teacher explain it that basically, for chromatic instruments, a lot of the notes (in terms of their frequency) are averages across all the scales. I spent quite a lot of time wondering how I might construct a diatonic guitar after that! –  overslacked Apr 26 '11 at 19:10
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You might find this math post interesting. It explains some of the theory behind equal- and non equal-tempered scales. –  Michael Apr 27 '11 at 16:39
    
it's a shame the two highest voted answers only address the frequency distinction in certain temperaments and not the functional distinction –  James Tauber May 14 '11 at 22:39
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Not an answer, but check out truetemperament.com for ways in which this can be used. –  Dr Mayhem Oct 15 '11 at 9:51

6 Answers 6

up vote 43 down vote accepted

See the section on tuning systems on Wikipedia for some background.

In short, most intervals do not sound best on equally-tempered scales (where the distance between any two consecutive half steps is the same) but on ones where the notes vary in distance. For instance, fifths usually sound the most in tune when the frequencies are in a 2:3 ratio.

Because of this, G♭ and F♯ will often sound different depending on which scale they're being used in and which notes they are played with. As far as I know, G♭ is never higher than F♯, always lower (or perhaps the same, like on a piano).

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Also useful is the Wikipedia article on comma –  Willie Wong Apr 26 '11 at 23:46
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there is a fantastic book "how equal temperament ruined harmony (and why you should care)" by Ross W. Duffin that approaches this answer in great detail. great answer –  Stephen Aug 19 '11 at 0:49
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Actually, to make them sound more 'in tune' harmonically (i.e. sounding at the same time with pitches of other names), G♭ CAN BE higher than F♯ : look for them near the lower left or lower right corners of this diagram en.wikipedia.org/wiki/File:Archicembalo_en_Cents.jpg "f# 579" below "♭G 620" in en.wikipedia.org/wiki/Archicembalo –  user1217 Sep 19 '11 at 10:20

It depends on the tuning system being used. If you're tuning by perfect intervals, i.e. intervals in which the ratios of the frequencies are in whole-number pairs, then Gb isn't exactly the same as F#.

For example, say you're tuning to A440 and using perfect intervals. Then the E above the A is tuned to 440 * 3/2 = 660 Hz. The B above the E is tuned to 660 * 3/2 = 990 Hz. The F♯ above the B is tuned to 990 * 3/2 = 1485 Hz. Meanwhile, the D below the A440 is tuned to 440 * 2/3 = 293.333 Hz, the G below the D is tuned to 293.333 * 2/3 = 195.555 Hz, and so on.

In the end, and adjusting for octaves, you get that Gb = 366.25 Hz while F# = 371.25 Hz. Not exactly the same, but pretty close. Not close enough not to be noticeable, though.

In Equal Temperament, pitches aren't determined by whole-number ratios. Rather, you use the formula:

frequency = 440 * 2^(n/12)

where n is the number of half-steps above or below the A440 reference pitch. This ensures that enharmonic equivalent notes have the same frequencies, but it also means that no interval is "perfect" in the whole-number ratio sense.

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Your formula is wrong. It should be 2^(n/12), and not 2^(log_12 n) It is easy to verify, because each octave should be 2*base note, so you should get 4 times the base for 24 pitch difference, but you get 2^(log_12(24)) = 2^(log_12(12) + log_12(2)) = 2*2^log_12(2) which isn't 4. –  SurDin Apr 26 '11 at 19:40
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@SurDin: Cripes, I'm having a bad day on the 'ole Stack Exchange. Of course you're right. Thanks for the correction. –  Alex Basson Apr 26 '11 at 19:45

There are the tuning differences, as already mentioned.

Then there is the function difference. If you have an entire piece in D major, using the tones in D major, seeing a D♭ instead of a C♯ would be very awkward.

When writing music, the rules (simplified) are:

  • use the tones of the key currently in use (could be a different key than the main key, the key of this specific beat, for example)
  • prefer ♯ when chromatically ascending, and ♭ when chromatically descending (this eases reading)

Bad spelling of enharmonies is something I see more and more. Probably because music publishing has become more and more easy so less and less skilled people do it, and because many think that a program's auto-transpose function actually works (or they know it doesn't, but they don't care).

http://www.musicarrangers.com/star-theory/p17.htm

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Professional programs such as Sibelius have numerous options to rewrite enharmonics according to many criteria (in facts I know at least 10 different plugins to Sibelius for that), and they are quite good at it, especially when dealing with orchestra music. The problem is further complicated by transposing instruments. –  ogerard Apr 26 '11 at 20:20
    
Sorry if I was unclear, transposing instruments is what I was referring to. It feels like people write everything in concert, then auto-transpose the winds that need it and press print. I have only an old version of Sibelius, but it was definitely not so smart then. –  Gauthier Apr 26 '11 at 20:24
    
Sibelius default transposing functions are better than before, what has improved much are the display options. But you really need the plugins written by Bob Zawalich to do professional work and default proof-reading. –  ogerard Apr 26 '11 at 20:34
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Also worth noting: one spelling might be better read vertically while the other better horizontally. For example in a choir piece featuring an F# major triad, the voice singing the third might prefer seeing a Bb, while it should functionally be a A#. I wonder if there is some kind of best practice in these cases. If the parts are printed separately I would prefer the horizontally readable solution. In a choir print where all singers have all parts, I'm not so sure. –  Gauthier Apr 27 '11 at 8:58
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@James: say you have a bar with two chords in it: B7 and F7alt. You play the second violin and start with the third of B7, slurred to the seventh of F7alt. Would you really have D# slurred to Eb? Of course you could rename B7 to Cb7, but would you really do that? –  Gauthier Jun 2 '11 at 9:35

When you are playing fretless string instruments, especially bowed instruments in small groups, you become very sensitive to these differences. I will not quantify them as there are already other answers on this area. When I was young, I was told the comma model of the occidental scale and I think it is a good first approach of these issues in most classical music, even if it is theoretical and limited. It gives you an easy way to remember the relative placements of accidentals.

Musically there are at least three musical dimensions where it is felt, one I would call melodic, another harmonic and still another timbral for lack of more nuanced words.

When you have a melodic lines that use accidental alterations, or hesitate between several modes and tonalities (this is quite common), the exact sensation you produce with your instruments is quite dependent of the exact intonation. There is a rhetoric quality to some music and it is enhanced by careful intonation. Depending on the period/school of the piece you may need to be very careful about the way you play accidentals. In fact certain qualities that are intuitively felt by listening to very good interpretation of music are due to a careful respect of these differences.

When playing double-stops (on a single instrument) or chords across a group, you cannot afford to play a Gb like a F# together with, say, a natural D. It is usually the same finger but not exactly the same place and angle of your finger on the fingerboard. It does not sound the same, but here again, chords are not isolated in a music piece. They should be heard in succession and only their contrast is really meaningful. As starting player usually make larger intonation errors than two or three commas, this is something which is usually treated after several years of study but could be intelligently made before.

The sound quality of your instrument can be different. A small change in pitch gives a different balance of resonance of the body of the instruments and of the other strings. Another direct aspect is that you cannot play for instance a Ebb on the D string of a cello. You have to find it by playing on the lower G string to be in tune. So it sometimes change fingerings. It is true with wind instruments such as the transverse flute where you have alternate fingerings to sound "more flat" or "more sharp" for certain notes in addition to the intonation changes that good player can create with their breathing technique.

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I've noticed this with the C#/Db on flute. Open fingering is a tad flatter than the pinky one, and I end up using them in different situations. –  Michael Apr 27 '11 at 16:46
    
@Michael: Yes. With wind instruments you have also "trill" fingerings which can be easier to use when switching rapidly but you would use non-trill fingerings for the ending note. –  ogerard May 2 '11 at 15:44
    
@Michael another C# fingering to explore is pressing two more keys - the two next to the pinky. –  8128 May 3 '11 at 19:57

As an isolated question, it's sometimes hard to understand why it's important that there is a difference but understanding why there is a difference is an important foundation to Western melody and harmony.

  • the vast majority of western music involves 12 notes in an octave
  • the vast majority of western music is based around a scale consisting of 7 of those notes specific to the choice of key (the notes are called the diatonic notes for that key)
  • a particular note in a piece is functioning either as a diatonic note or as a note a semitone higher or lower than a diatonic note
  • when expressing a note that is functioning as a raised or lowered note, you use the same letter name as the diatonic note you are raising or lowering. e.g. a raised G is G♯ and a lowered G is G♭.
  • if the diatonic note is already written with a sharp, the raised note has a double sharp and the lowered note has a natural symbol
  • if the diatonic note is already written with a flat, the raised note has a natural and the lowered note has a double flat
  • but in all cases, the letter part of the note name stays the same

So, imagine you're in the key of D major. The diatonic notes are: D E F♯ G A B C♯. What does F♯ mean? It means the third note of the scale. What does G♭ mean? It means you've taken the fourth note of the scale and lowered it it.

In 12-tone equal temperament, they may sound the same; you may play them the same on the piano or the guitar. But if the function of the note at a particular point in the piece is as the third note in the D major scale, you can only write it F♯ and not G♭. F♯ means something completely different.

It's the musical equivalent of "hear" versus "here". Just because they are homophonic doesn't mean they are the same word. Similarly, in western tonal music F♯ doesn't mean the same as G♭.

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A term to research is 'temperament'. There are a number of books on the subject. In much-too-short, on fretless instruments, performers can choose to tune intervals as described in another answer to meet their artistic sense, and in some cases end up with the enharmonics distinct. On pianos and keyed and fretted instruments, someone has to make a decision. For hundreds of years, the standard decision has been equal temperament. Further, when an ensemble plays in tune, even the fretless gang has to play along.

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