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I was just reading the Wikipedia page on the note F (as I do every evening) and was confused by this part where it says that even though F♮ and E♯ are enharmonic they “do not sound the same”:

E♯ is a common enharmonic equivalent of F, but is not regarded as the same note. E♯ is commonly found before F♯ in the same measure in pieces where F♯ is in the key signature, in order to represent a diatonic, rather than a chromatic semitone; writing an F♮ with a following F♯ is regarded as a chromatic alteration of one scale degree (E♯ and F♮ do not sound the same, except in some tunings that define the notes in that way).

What does the author of this sentence mean? Do they not by definition sound the same?

42

The thing is that the "some tunings that define the notes in that way" in the Wikipedia quote include the most common tuning today, 12-tone equal temperament (12-TET). So, E# and F natural do usually sound the same.

...But not always. Change the tuning system and you can easily have an E# and an F natural that sound slightly different. Just intonation will likely do it, since its perfect fifths are slightly larger than 12-TET's. (Just intonation is a mess the more of the chromatic scale you want to tune with it.)

  • 1
    So do you mean that it’s referring to microtonal music when it says some tunings? – Aran G Dec 26 '18 at 19:33
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    @AranG It's referring to microtonal music, and also the different tunings if they don't count as microtonal. – Dekkadeci Dec 26 '18 at 19:37
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    If you have a system that defines E# and F as different frequencies, then that is not a 12-tone system. In any 12-tone system, E# and F are the same pitch class. What can happen is that you can change your tuning system on the fly if the instrument allows tuning adjustments. But "E# and F do not sound the same" is misleading, if not outright incorrect. – MattPutnam Dec 27 '18 at 1:26
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    @Dekkadeci - just intonation isn't mictotonal. It merely uses all the notes with slightly different tunings. Microtonal splits notes we are used to into more parts. I guess you may mean 'microtonal' to encompass say, an unfretted instrument that can make E# in one key slightly different from F in another? – Tim Dec 27 '18 at 9:57
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    Seems like poor phrasing on Wikipedia's part then, acting like the case is a rare exception when it's actually "the most common tuning." – Kevin Dec 29 '18 at 17:25
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I think this particular phrasing is rather confusing, as it is trying to talk about two concepts at the same time: enharmonic equivalence, and intonation.

The concept of intonation (and temperament, which relates to systems of intonation) deals with the fact that even given a certain reference pitch (such as A4=440), there is no one absolutely correct frequency for the other notes to be sounded at. The exact frequencies of notes might be selected to make a certain key sound harmonious, or to be a good compromise that allows a range of keys to sound good (such as 12-tone equal temperament).

On instruments that allow the intonation to be varied by the player (such as fretless stringed instruments), the very same note - even with the same name - might be sounded at a slightly different pitch to make it sound better in a certain chord or melodic phrase. So even two notes notated as E4 might not be at the same pitch; following the logic in the quote from Wikipedia, one could go so far as to say "E and E do not sound the same".

So when the article says "E♯ and F♮ do not sound the same, except in some tunings that define the notes in that way", the fact that the note might be called both 'E♯' and 'F♮' is a little bit of a red herring; a note's intonation might vary regardless of variations in how it is named. Nevertheless, there might be some contexts in which the note notated 'F♮' tends towards one pitch, and 'E#' tends towards another.

5

Some tunings are designed so that, whenever possible, two notes which are separated by a perfect fifth will have a precise 3:2 frequency ratio.

If that 3:2 relationship holds between A#->E#, then D#->A#, G#->D#, C#->G#, F#->C#, and B->F#, that would suggest that the frequency ratio between B and the E# above it would be 729:512 (about 1.42).

On the other hand, if that 3:2 relationship holds between F and C, C and G, G and D, D and A, A and E, and E and B, then the frequency relationship between the B and the F above it would be 1024:729 (about 1.40).

It would be possible for all the 3:2 relationships to hold if E# and F were recognized as different notes with slightly different pitches, but if E# and F are the same pitch then at least one of the perfect-fifths relationships much involve something other than a perfect 3:2 frequency ratio.

  • Which tunings are designed that way? – phoog Dec 31 '18 at 7:25
  • @phoog - I believe Pythagorean tuning is. – Dekkadeci Dec 31 '18 at 10:07
  • @Dekkadeci true, but you can't use it very well for harmonic music, and nobody actually does. – phoog Dec 31 '18 at 16:47
  • @phoog: There is only one possible tuning that uses a precise 3:2 ratio for all perfect fifths, since locking in that ratio for all fifths would precisely define the mathematical relationship among all the notes. Most tunings make some or all of the fifths and/or octaves "slightly imperfect" to minimize or eliminate the distance between pitches that are a diminished second apart, but if one is using a tuning where pitches can be a diminished second apart have slightly-different frequencies, it's important to have notation that can distinguish them. – supercat Dec 31 '18 at 18:31
  • @supercat for a twelve-tone keyboard, the maximum number of pitch classes that can be in 3:2 ratio is eleven. If you tune the keyboard that way, the twelfth interval of seven half steps will be in the ratio of 262144:177147. Even ignoring that "wolf," the thirds in such a system are dreadful, which is why I said it is not useful for harmonic music. But I've found that it's not necessary to have such a notation as you describe to achieve just intonation, just a good pair of ears and adequate rehearsal. – phoog Dec 31 '18 at 22:46
2

If you know the physics as well as the aesthetics of music it helps. Here it would take too long to cover all of this however here's a start.

Suppose an amateur wanted to tune a piano and all they had was a tuning fork. For simplicity let's say it sounds middle C.

The amateur who has an excellent musical ear but has not undergone a year's training as a piano tuner, proceeds as follows:

(1) Tune middle C on the piano to the tuning fork

(2) Tune all the other Cs on the keyboard to be perfect octaves from middle C. So far so good but what to do next? Let's continue as follows.

(3) The next 'purest' interval after an octave is the perfect 5th. So tune all the Gs on the piano by ear to sound perfectly in tune with the Cs. Everything sounds great.

(4) Assuming we have all the Gs in tune we can go up another 5th to D, excellent.

(5) Go from D up a perfect 5th to A

(6) Continue the process, A to E, E to B, B to F#, F# to C#, C# to G#, G# to D#, D# to A#, A# to E# (which you might be tempted to call F but let's not), E# to B#. Now we're on B# so hurray! we'are back to C because "B# and C are the same" - yay you have completed the circle of 5ths.

So now you have tuned every single note on the piano simply by octaves and perfect 5ths.

Present your work to a pianist who sits down to play. They will produce the most appalling racket that you, they or anyone else has ever heard. The result will be slightly less unpleasant if they play simple tunes in C major but the key of F# will be completely unlistenable.

Why? Because of the mathematics. If you go up in 5ths indefinitely you will actually never end up perfectly in tune no matter how many times you go round the circle of 5ths. This has to do with logarithms so if you don't like maths don't pursue that line of enquiry.

There are other threads that go into more detail, e.g. Why is the perfect fifth the nicest interval?

  • 2
    "Why? Because of the mathematics." This does not seem like an attempt to answer the original question. – sean Dec 28 '18 at 13:18
  • @sean You're right. I got called away to deal with something in real life. There is more to it but I'll have to find time to continue with it. However by indicating that this system of tuning does in fact produce a B# that does not equal C (and also an E# that does not equal F), I think I have at least made a start. A 21st century piano tuner definitely does not use this method but instead uses equal temperament which is a kind of fudge. It also cause problems when a piano accompanies a violin for instance. The pianist can't adapt so the violinist has to - and not every violinist knows that – chasly from UK Dec 28 '18 at 13:44
  • The impossibility of closing the circle of fifths has more to do with the fundamental theorem of arithmetic than with logarithms. Each time you go up by a fifth you add a factor of three to the numerator, and there's no way to remove it through division by two. And if you want consonant thirds, you've got problems before you even start worrying about completing the circle. – phoog Jan 1 at 22:26
  • That's not true. If you go up in fifths you multiply the frequency by 1.33333... each time. This explains why the equally spaced strings on a grand piano have the appearance of a logarithmic graph (apart from the very lowest notes which would make the instrument unwieldy) and with a correction for the mass per unit length of the string. Consonance of other intervals under this (ineffective) scheme of tuning doesn't come into it. The supposed thirds will eventually appear if you naively tune by going up in fifths but their pitch will be fixed relative to where you started. – chasly from UK Jan 1 at 22:54
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In the harmonic series (fundamentals and overtones on a single string or a wind/brass instrument), the the harmonics are at 2x, 3x, 4x, 5x ... the original frequency. The harmonics of C are approximately:

C1 C2 G2 C3 E3 G3 Bb3 C4 ...

(aside: the Bb is particularly badly "out of tune" - that's why a piano has the hammers aligned to strike the string where they do... to avoid exciting the 7th harmonic).

This is why C and G make a good chord: because C1,G1 are at a ratio of 3/2.

BUT... equal temperament means that all semitones must be the same on a log scale... There are 7 semitones in a perfect 5th, and 12 in an octave. So a perfect 5th is defined ALSO as being a factor of 2^(7/12). Which is almost, but not identical to 1.5.

Thus the requirement of 2x octaves, and perfect-5ths is not compatible with equal temperament. [This difference is the "Pythagorean comma")

On a piano, it's a bodge (5ths aren't actually that far out, but major 3rds are much flatter than they "should" be). Good singers can adjust their tuning according to key - this is one reason why "dissonant" music sounds much sweeter when sung than played.

  • Equal tempered major thirds are much sharper than just major thirds, not flatter. To have a just major third, it must be considerably lower than the pitch on the piano, not higher. To be specific, a just major third is in the ratio 5:4, or 1.25, while an equal major third is the cube root of two, or 1.259921. That makes the just C# above A440 550 Hz while the equal C# is 554.365 Hz. – phoog Dec 31 '18 at 23:29
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Totally disagree. This paragraph is not about whether the two notes sound the same melodically, but whether they sound the same harmonically. Depending on key and counterpoint there are times when it is clearer to label a note Fnatural instead of Esharp. This also leads to double flats, double sharps, etc. The end result is purely academic, but makes compositional intent clearer to people who are well versed on the academics. The big hint here are the terms diatonic, chromatic, and key signature which have little or no meaning in atonal music.

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    I think you missed the term "chromatic semitone" in the quote, along with the implied "diatonic semitone". According to en.wikipedia.org/wiki/Semitone at the time of this comment, the two semitone types may be of different sizes. – Dekkadeci Dec 27 '18 at 15:09
  • Not at all. In fact it is key to my argument. Every key has a single diatonic note of every letter A-G. You can’t have Fnatural and Fsharp both as diatonic notes in the same key. So it even though we typically think of Esharp as Fnatural (an artifact of basing our musical language around the the key of C) it is not always the correct way to name it. They key of Fsharp has an Esharp as it’s 7th degree, not F. – Garrett Berneche Dec 27 '18 at 15:51
  • It is correct to say that on an instrument perfectly tuned to the key of Fsharp compared to a instrument that is perfectly tuned to the key of Fnatural the (for the sake of argument we will assume a keyboard instrument) the F key would not produce the same pitch on both instruments, but you would not use the term semitone to describe the difference. – Garrett Berneche Dec 27 '18 at 17:29
  • If the author did indeed mean to speak of microtonal differences then they changed definitions and subjects in the middle of a paragraph. Bad form! I have to assume, based on syntax, they did not mean to do any such thing. – Garrett Berneche Dec 27 '18 at 17:47
  • @GarrettBerneche what do you mean by "perfectly tuned to the key of..."? If you mean so that all intervals are just, there's no such thing. It's impossible to tune a keyboard to any one key. Also, given Wikipedia's model, it is likely that the paragraph was written by multiple authors at different times. Changing subjects in the middle of a paragraph is rather likely. – phoog Dec 31 '18 at 7:21
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On an instrument with fixed intonation like a piano or organ, E sharp and F natural are the same frequency. However with an instrument like the voice or a violin which can potentially produce a sound at any frequency in their tessitura, E sharp and F natural can be interpreted differently from each other by the performer, depending on the context.

For example, on an equally tempered scale - like on a piano, C would be 523.25 Hz and C♯ would be 554.37 Hz. If the context of E♯ (sung for instance) is that it is a perfect major 3rd (frequency ratio of 5/4 - fifth harmonic, two octaves lower) above the tonic of C♯, that frequency would be 692.96 Hz. If the context of F is that it is the sub-dominant of the tonic C (frequency ratio of 4/3), that frequency would be 697.67 Hz.

protected by Dom Jan 2 at 16:12

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