Concert pitch and middle C

[leftybassman392]

Regulars of this parish will know that I like to play around with numbers.

I quite often run random number things around in my head as I'm waking up. (No, I don't think it's weird at all but thanks for asking. )

This morning I got to thinking about the relationship between concert pitch (A4, commonly standardised at 440Hz) and Middle C (C4, often thought of as being at 256Hz but historically it's a moveable feast, and in truth comes about for slightly different reasons). What follows is a bit of an exploration of the numbers involved and is a bit theoretical. When it comes to tuning actual instruments of course, things can get a bit fiddly (as any piano tuner will be happy to tell you).

Most modern instruments are tuned using a fairly complicated mathematical system called Equal Temperament. If you use 440Hz as the standard (not everybody does - more in a mo) you can work out the frequency of every note in the audible spectrum. The great advantage of this system is that it allows you to do all sorts of clever stuff like modulation and transposition that are much more of a challenge in traditional tuning systems. The downside is that it's mostly a teensy bit off-pitch - not a lot mind, and in truth we're so used to using it that we generally don't notice anything wrong. Electronic tuners that show cents use this system.

In past centuries, scales were assembled using a variety of systems based on the natural harmonic series (2:1, 3:1, 4:1,...). The earliest, commonest and most easily understood method was what most will know as cycle-of-fifths tuning. (Historically it's actually more accurate to call it Pythagorean tuning, but I digress). The upside is that the pitch relationships are absolutely bang on harmonically (which makes it popular with specialists in medieval and early renaissance music), but the downside is that if you try to play in a different key without retuning your instrument and/or adjusting your fingering it can sound catastrophically out-of-tune.

I'll be happy to demonstrate these methods at work if anybody wants me to, but for now we'll keep things simple.

 

So, Concert Pitch. The internationally accepted standard is A4 = 440Hz. Not everybody uses this though, and it is possible to find people using a standard anywhere from around 420Hz up to around 460Hz. (This isn't the same as detuning by the way. When you detune a guitar or bass, you're raising or (more commonly) lowering the pitch of your instrument in multiples of a semitone relative to whatever your standard pitch is.)

Using 440Hz for A4 and using an Equal Temperament calculation gives middle C at around 262Hz. This is about as close as you can get to a standard for this note using Equal Temperament.

There's been a bit of a thing recently about setting A4 at 432Hz. People who do this like to talk about the sound being slightly darker. Personally I'm not convinced - the only way you can tell the difference is by playing them directly back-to-back (which of course nobody in their right mind would do in a performance situation). At the risk of being slightly controversial, it strikes me as being a bit of trendy oneupmanship.

What is more interesting is the Maths that sits behind it:

If you set A4 at 432Hz, Equal Temperament gives middle C as a whisker over 256Hz.

More interestingly still, if you work from middle C at 256 Hz and go in the other direction using an old-fashioned Pythagorean sequence (A4 is a major 6th above middle C4 with a ratio of 27/16) you get A4 at exactly 432Hz.

Kind of suggests to me a bit of putting the Pythagorian cart before the Equal Temperament horse. Except of course that 256Hz is by no means universally accepted as the standard for middle C in the first place...

 

As I said up front though, I'm just playing around with numbers.

 

Edited May 11, 2019 by leftybassman392

[mcnach]

1 hour ago, leftybassman392 said:

 

I'll be happy to demonstrate these methods at work if anybody wants me to, but for now we'll keep things simple.

 

 

 

If you have the time, and the will, yes please  Very interesting!

[MacDaddy]

Hence Bach's Well Tempered Clavier. 

[leftybassman392]

1 hour ago, mcnach said:

 

If you have the time, and the will, yes please  Very interesting!

No sooner said than done...

Before I start, I should add that there's a series of articles I did that now resides at the dusty end of the 'Theory and Technique' shelf and which goes into this in a bit more depth.

Equal Temperament:

F_t - target frequency

F_r - reference frequency (440Hz assumed but works for any reference frequency)

n - number of semitones between the pitch of the target note and the pitch of the reference note

Algorithm: Ft = Fr x 2^(n/12).

Example: F#5 (9 semitones above reference) = 440 x 2^(9/12) = 740.0 Hz

Pythagorian algorithm:

F_r - as above

Ft - successive applications of 3/2 multiplier until required pitch is achieved, doubling or halving as required to keep result within desired range

Example: F#5 (maj 6th above A4) = 440 x 3/2 x 3/4{3/2 and halve} x 3/2 = 440 x 27/16 = 742.5 Hz

[Procedural note: Pythagorean algorithm sequence gives A>E>B>F# . B value dropped an octave]

 

 

Edited May 11, 2019 by leftybassman392

[leftybassman392]

On 11/05/2019 at 11:49, MacDaddy said:

Hence Bach's Well Tempered Clavier. 

Indeed, with the caveat that Well Temperament is essentially a fine-tuning of the existing harmonic intervals, whereas Equal Temperament is a very precise mathematical process. Whilst the end result is similar (i.e. keeping good(ish) harmonic integrity while allowing modulation), the process is somewhat different.

Edited May 12, 2019 by leftybassman392

[leftybassman392]

Quick update: I should also mention that for notes below the reference in Equal Temperament the power of 2 in the formula is negative.

Example:

To get the F# that is 3 semitones below the reference and 1 octave below F#4, the formula would be

F#3 = 440 x 2^(-3/12) = 370.0Hz

 

 

For the Pythagorean sequence, simply divide the earlier result by 2 to show the dropped octave:

440 x 27/16 x 1/2 = 440 x 27/32 = 371.25Hz

For (sort of) completeness, the ratios in the Pythagorean Major scale are:

9/8, 81/64, 4/3, 3/2, 27/16, 243/128 and 2/1.

(I've left out the chromatic notes for reasons of simplicity and also because it means I don't have to get into Pythagorean commas and wolf fifths. Phew!)

 

Should have put this into the earlier post. Sorry.

Edited May 12, 2019 by leftybassman392

[SpondonBassed]

Thanks for this but excuse me... my brain Hertz.

[tauzero]

There's a book on the subject that is quite interesting: https://www.penguinrandomhouse.com/books/85560/temperament-by-stuart-isacoff/9780375703300/

[leftybassman392]

10 hours ago, tauzero said:

There's a book on the subject that is quite interesting: https://www.penguinrandomhouse.com/books/85560/temperament-by-stuart-isacoff/9780375703300/

Interesting. It's not a title I'm familiar with, but the ideas (if not all the names) in the synopsis seem pretty familiar.

Thanks for posting.

 

While I'm here, there's a fairly good youtube vid that covers this information (actually there's a few, but this one is about as down-to-earth as the subject allows):

 

Edited May 13, 2019 by leftybassman392

[fretmeister]