Why are C♯ and D♭ different frequencies?

Question

I am a music enthusiast, and I was recently reading What is the difference between equivalent Flat and Sharp keys as far as musical notation? Are there any reasons to prefer one over the other?

This part struck me as odd:

C♯ and D♭ actually differ 41 by cents from each other

As far as I know, there should be 2 semitones between C and D. Moreover, C♯ is one semitone above C and D♭ is one semitone below D. Therefore, C♯ and D♭ should be equivalent. If so, how can C♯ and D♭ actually differ by 41 cents from each other?

Answer 1

The linked answer is a bit of a mess, and it's a common mess for people to make.

When we talk about the exact frequencies of each pitch class, we have to know the temperament, and a reference pitch. For example, 12-tone equal temperament (12TET) with A4=440Hz is a standard in modern music. From those two parameters, we can extrapolate the exact frequency of every possible note.

12TET is nearly ubiquitous nowadays (at least in Western music), but it doesn't sound as clean as Just Intonation (JI). In essence, 12TET has made every key sound equally imperfect. JI creates a scale where the intervals in the primary chords are all very nice simple ratios, and so the chords ring out very cleanly, but it only works in that key. Important note: within a given JI tuning, each of the 12 pitch classes still has only one single frequency. There is no difference between C♯ and D♭ in, say, "Pythagorean tuning based on A, with A=440Hz".

But most music doesn't stay in one key. While a piano can't make pitch adjustments on the fly (which is why we've agreed to use 12TET for it), most instruments in an orchestra can. So when the piece is in A major, the orchestra will use JI and adjust C♯ to be a little flatter than it would be if using 12TET. But then if the piece modulates to F♯ minor, they'll start playing it slightly sharp.

When people say that C♯ is not the same as D♭, what they really mean (whether they realize it or not) is that context may make for different micro-adjustments. In C major, a C♯ might be the third of an A major chord, perhaps a secondary dominant of the ii chord, while D♭ might be the root of the Neapolitan chord. These would result in different tuning choices.

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(edited from comment suggestions, some comments are now orphaned)

Answer 2

The short answer is that for 12-tone equal temperament (12TET), the de facto tuning system for western music, Db and C# are exactly the same sounding note. Exactly what frequency that note sounds like for a given octave also depends on the pitch reference, which is typically A4=440Hz.

According to 12TET, we break the octave into 12 equal ratios. Since an octave is a ratio of 2:1, the ratio from one note f1 to the note 1 semitone higher f2, is calculated as f2 = f1*2^(1/12) with 2^(1/12) ~= 1.059463.

While this is by far the most common tuning system you will encounter (in a western context at least), it is only one tuning approach, and is relatively modern compared to many alternatives you may encounter, including the Pythagorean system mentioned in the question you referenced (which as its namesake suggests, is thousands of years old).

The Pythagorean tuning system takes the approach of determining each note by calculating the perfect fifth using the ratio of 3:2, or 1.5 times the reference frequency. Apart from being a simple ratio, this tuning system is actually very easy to implement because that exact frequency (strictly 3:1, an octave up from 3:2) will already be present in the harmonic series of the reference note for most music instruments (string and wind instruments including the human voice). This is certainly the case for violinists, who tune their strings (which are perfect fifths apart) by this method.

However, a perfect fifth under Pythagorean tuning is approximately 702 cents, as opposed to exactly 700 cents in 12TET. If you continue tuning this way forever you will never reach the same pitch again. As you tune around the circle of fifths, you will build up fractions with bigger powers of three 3^n over bigger powers of two 2^m and there is no way that fraction will ever equal 1 (the reference pitch) except when m = n = 0, ie the reference pitch you started with.

If we calculate the ratios from G (since G is the furthest pitch from C#/Db in both directions), going up in fifths would look like:

G -> D (3/2) -> A (9/4) -> E (27/8) -> B (81/16) -> F# (243/32) -> C# (729/64)

If we go back the other way (that is, down by perfect fifths), it looks like this:

G -> C (2/3) -> F (4/9) -> Bb (8/27) -> Eb (16/81) -> Ab (32/243) -> Db (64/729)

If we normalise the resulting fractions so that they occur within the same octave, it works out to be C# at 729/1024 ~= 0.71191 vs Db at 512/729 ~= 0.70233, which will obviously sound different. I calculated the difference between these notes at 23.46 cents, not the 41 cents mentioned in the referenced question.

To put these numbers into perspective, if we assume that A is 440Hz, then we can determine the reference G as being two perfect fifths away at 8/9 x 440 or ~391.11Hz. Using this G, we can find the Pythagorean Db and C# directly below that G using the ratios above at ~274.689Hz and ~278.436Hz respectively. Compare this to 12TET with A4=440Hz, we would have G just below at ~391.995Hz and the enharmonic Db/C# at ~277.183Hz.

It is unlikely that you will encounter a situation in which C# and Db actually sound even 23.46 cents apart for a number of reasons. The first and most obvious reason is that 12TET is ubiquitous in western musical contexts. Most modern fretted instruments (guitars/basses) and keyboard instruments (piano, organ, etc.) are tuned according to 12TET.

Even in the rare case that you have a collection of vocalists performing a cappella, such as in a barbershop quartet, they likely won't drift too far away from conventional tuning thanks to tonal memory. Basically, even people without perfect pitch can have some memory of pitches such that the more 'natural' tuning systems, such as Pythagorean, will be modified by their memory of the 12TET pitches they have probably heard their whole life.

Answer 3

As already said,

• The post you asked about refers specifically to C♯ and D♭ in Pythagorean tuning.
• The discrepancy of 41 ct is wrong, no idea how that came about^{See below}.
• Pythagorean tuning is only one of multiple just-intonation systems.

So in fact, not only are C♯ and D♭ different notes, there are actually multiple different notes you could call C♯! To give a better idea of the different options, here is an overview over how these notes can be constructed in the different tuning systems using integer frequency ratios, always starting from C, and how the results compare to 12-edo.

http://nbviewer.jupyter.org/gist/leftaroundabout/b0708139f867a160579f14c0b04caeb8

Pythagorean, upwards

Constructed only from pure fifths upwards and fourths downwards (or equivalently, only fifths upwards with octave-compensation).

onKeyboard $ constructNote PreferSharps [3/2, 3/4, 3/2, 3/4, 3/2, 3/4, 3/4]

Pythagorean, downwards

Fifths downwards and fourths upwards.

onKeyboard $ constructNote PreferFlats [4/3, 4/3, 2/3, 4/3, 2/3]

So you see, this D♭ is 24ct flatter than the Pythagorean C♯.

Ptolemaic, upwards

Constructed from fifths and just major thirds upwards / fourths downwards.

onKeyboard $ constructNote PreferSharps [3/2, 3/4, 5/4, 3/4]

Note that this is flatter than the 12-edo pitch. In fact it's much closer to the Pythagorean D♭ than to the Pythagorean C♯!

There's an alternative construction which comes out yet a lot flatter:

onKeyboard $ constructNote PreferSharps [4/3, 5/4, 5/8]

This is quite extreme, I doubt any classical musician would ever play C♯ that low. But here, as Mr Lister pointed out in the comments, we appear to have found the 41ct from Dorien's answer, namely if we compare this C♯ to the next option for D♭:

Ptolemaic, downwards

Here, we reach D♭ very quickly, after only a fourth up and major third down:

onKeyboard $ constructNote PreferFlats [4/3, 4/5]

So what the hell, you may well ask at this point. What's the correct version now?

Well, it depends on the context! But although this is often claimed – for classical Western music, Pythagorean tuning is not very relevant. This music makes heavy use of harmonies based around major chords, and major chords only render sensible in Ptolemaic tuning, namely as ratios 4:5:6, compared to Pythagorean's 64:81:96. (Nobody can actually distinguish frequency ratios with such high numbers by ear!)

Thus you can as a rule of thumb say that C♯ is a bit flatter than D♭. The literature confirms this, e.g. Leopold Mozart:

...alle durch das (♭) erniedrigten Töne um ein Komma höher als die durch das (♯) erhöhten Noten. Z.B. Des ist höher als Cis; As höher as Gis, Ges höher als Fis u.s.w.

Translation:

All tones that are lowered with (♭) are a comma higher than the (♯)-raised notes. E.g. D♭ is higher than C♯; A♭ higher than G♯, G♭ higher than F♯ etc..

He also adds

Hier muss das gute Gehör Richter sein

Here, the good hearing sense should judge

In other words: there is no single rule one can apply to deduce the perfect frequency for any given named tone, one should always listen carefully what actually sounds best.

Answer 4

The first thing to understand is that if you want to go up by a constant interval, you multiply the frequency by a particular number.

For example, to go up by an octave, you multiply the frequency by 2. Since multiplication by 2 is the simplest multiplication we can do, this sounds pleasing to the human ear - so pleasing, in fact, that we learn to hear the two notes as the same.

If we want to go up by two octaves, we multiply by 2 again, for a combined total of 4 times the original frequency. And so on.

But there are other nice numbers that we can multiply the frequency by. If we multiply by 3, for example, then we go up by an octave and a fifth. To get a fifth, we go back down the octave by dividing by 2, so a fifth corresponds to multiplying by a factor of 3/2.

If we multiply by 5, then we go up by two octaves and a major third. So a third corresponds to multiplying the frequency by a factor of 5/4.

Thirds, fifths and octaves are fundamental to Western music, and all other intervals are built from them. The reason that they sound so nice and concordant is because they are built up from very simple multiplications.

For example, if we start at C and multiply by 5/4, we get to E, and if we multiply again by 5/4 we go up another third to G♯. Now if we divide by 3/2 to go down by a fifth, we get to C♯. The total multiplier is

5/4 * 5/4 * 2/3 = 25/24 = 1.041666...

If instead we multiply by 2, we go up to a high C. Now, if we divide by 3/2, we go down a fifth to F. If we now divide by 5/4, we go down by a third to D♭. The total multiplier is

2 * 2/3 * 4/5 = 16/15 = 1.06666...

Since these two numbers are so similar, it's easy to get confused between the notes C♯ and D♭.

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'Now, hang on!' I hear you say. 'C♯ and D♭ aren't just similar notes - they are the same note! After all, they both occupy the same key on my piano keyboard!'

This is actually a very clever musical trick. In order for piano keyboards to make sense, they can't treat C♯ and D♭ as separate notes, at least not if they want to avoid something horrific like this:

this is known as a split-key keyboard, of the type used in the 16th century when they were still figuring this stuff out

Instead, we need to approximate notes so that we can make a scale using only twelve different tones. So we end up having one key for both C♯ and D♭. Pressing this key might play a C♯, it might play a D♭ or it might play something in between.

A choice of approximations is called a temperament, and there were many different temperaments used right up to the Classical period. The title of J. S. Bach's 'The Well-Tempered Clavier' refers to one such temperament.

Different musicians had different preferred temperaments. One common quality was that certain keys (normally 'white-note' keys, such as C major) would sound very pure and concordant, while others would sound more off-key and spicy. This was sometimes considered a desirable feature of a temperament: different keys had different characters.

The temperament used almost universally on modern pianos is much more boring, but also more versatile. It is called 'Equal Temperament', and its name means that all of the semitones on the keyboard are exactly the same interval apart. An equal-temperament semitone is exactly a 12th of an octave, so it corresponds to multiplying the frequency by

the twelfth root of 2 = 1.05946309436....

(notice how this comes in between the 1.041666 and 1.0666 that we calculated earlier!)

Now, what does an equal-temperament fifth sound like? Well, it sounds like the twelfth root of 2 raised to the seventh power (since there are seven semitones in a perfect fifth):

2 ^ (7 / 12) = 1.49830707688...

By a brilliant mathematical coincidence, this is almost exactly equal to 3/2. So there is no audible difference between a fifth on a piano (1.498...) and a fifth that you would naturally sing (1.5).

What about the major third? A major third is four semitones, which corresponds to

2 ^ (4 / 12) = 1.2599...

This is still fairly close to 5/4 = 1.25, but now the difference is audible (there are some sound recordings on https://en.wikipedia.org/wiki/Major_third that you can listen to). A major third on a piano is noticeably different from a major third that you would naturally sing.

For the most part, you don't have to worry too much about this when you are making music, but it's worth keeping in mind sometimes.

Answer 5

There's pure tuning, where intervals are in simple frequency ratios, following the harmonic series. It gives very beautiful chords, but only in one key. Change key, you have to re-calibrate. And sudden CHANGES of key, which today's music do a lot, can sound a bit odd. So there's a compromise system, equal temperment, where all semitones are equal. It's never quite right, but it isn't TOO wrong, and our ears have got used to it. That's what a piano uses. It has to, really!

Answer 6

The key phrase in that answer that you missed was "In Pythagorean tuning…". As the Wikipedia article says,

The so-called "Pythagorean tuning" was used by musicians up to the beginning of the 16th century. "The Pythagorean system would appear to be ideal because of the purity of the fifths, but other intervals, particularly the major third, are so badly out of tune that major chords [may be considered] a dissonance."

Because of the wolf interval, this tuning is rarely used today, although it is thought to have been widespread.

Basically, the difference between C♯ and D♭ is mainly of historical and theoretical interest today. It is precisely because of inconvenient discrepancies like this 41-cent difference between enharmonics that nearly all modern music prefers other tuning systems.

Answer 7

John Gowers, in his answer, explained how the intervals C-C♯ and C-D♭ can have frequency ratios 25:24 and 16:15. 25:24 is ~70.67 cents and 16:15 is ~111.73 cents. The difference is 41.06 cents, thus vindicating the text quoted by the OP.

We should not assume Pythagorean tuning, that is, building all intervals from octaves and pure perfect fifths (frequency ratio 3:2). Pythagorean tuning is one possibility but it is not the only one available.

Still less should we assume 12ET in which the only possible intervals are multiples of a 100-cent semitone.

Answer 8

It depends on the tuning. In 31-TET, there are 5 steps of size 5 and 2 steps of size 3 in the C major scale. A sharp or flat raises a note by size 2. Consequently, C♯ is one 31th octave below D♭ which makes for 1200cent/31 = 38.7cent of difference. Well, almost there.

The original statement presumably talks about some form of Pythagorean or pure tuning but it is not clear just which scale and tuning are used for the statement.

Answer 9

a general consideration:

• every single enharmonic frequency has a different frequency from one another;

people just ignore that most of the cases, either:

• because of impracticality of making different pitches, for instance on a violin or on a flute or on a trumpet, or on a electric guitar, vocals whatever

simply because that's physically extremely difficult to be done. or:

• because in a instrument like a piano, for instance, usually the pitches are converged into the same key.

but theoretically every single enharmonic pitch, should have its one intonation, which depending on the note, should be:

• 5-10 cents maximum distance between one another;