Tuning, pitch question

[Bob]

This is sounding familiar.  I may have thought about or asked about this years ago.

I'm thinking in terms of exact pitch here.  Cents.  Hertz.  More like Hertz to have an exact number.

And say you take a standard tuner, no special temperament set.

Does each pitch have its own exact number of Hertz?  I would imagine it would.  Is that based off one note, like A or C?  Because if you start on one of those, say A, and then base C on that C, you end up with "Equal temperament C, based off A as the starting point."  But what if you started off G instead of A?  Do you end up with exactly the same C?  Is "Equal tempered C, based off A" the same as "Equal tempered C, based off G?"

I suppose A would make sense as "the" reference point.  A is 440.  But then would a note like G **always** be the exact same G, if the key were G?  If you've got A 440, is G always a certain number too like that?

And is that what a standard tuner is measuring?  A is 440.  G is always ____Hz.   F is always ____ Hz?  And that's based off equal temperament?

[brogers70]

Here's a table of the frequency of all the pitches in equal temperament with A440. If instead of calling it A440, you called it G392, you'd still end up with the same frequencies for each pitch.

https://pages.mtu.edu/~suits/notefreqs.html

[timothy42b]

However, you don't tune a piano to those calculated pitches, for a couple of reasons. 

One is that our ears don't perceive in a linear fashion across the whole range of a piano keyboard.  So in some ranges our ears want a pitch higher or lower than the calculated.  Stretch. 

Another and more important is that the piano string pitch is not one frequency, but a whole series of frequencies.  The overtones don't line up neatly (integer ratios) like they do on some wind instruments.  And this is slightly different on every piano.  So if you have a chord where the fundamental frequencies are correct but the overtones clash, your piano sounds bad.  Piano strings are steel and they have the same stiffness as a crescent wrench. 

So - and I'm not a tuner, so I might have something wrong - you don't really tune a piano to equal temperament, you tune a piano so it sounds as close to equal temperament to the human ear as you can get it. 

[brogers70]

However, you don't tune a piano to those calculated pitches, for a couple of reasons. 

One is that our ears don't perceive in a linear fashion across the whole range of a piano keyboard.  So in some ranges our ears want a pitch higher or lower than the calculated.  Stretch. 

Another and more important is that the piano string pitch is not one frequency, but a whole series of frequencies.  The overtones don't line up neatly (integer ratios) like they do on some wind instruments.  And this is slightly different on every piano.  So if you have a chord where the fundamental frequencies are correct but the overtones clash, your piano sounds bad.  Piano strings are steel and they have the same stiffness as a crescent wrench. 

So - and I'm not a tuner, so I might have something wrong - you don't really tune a piano to equal temperament, you tune a piano so it sounds as close to equal temperament to the human ear as you can get it.

I'm not sure I follow this. If the fundamental frequencies are correct (according to equal temperament) then the overtones have no choice but to clash. Tuners who tune by ear know that there will be beats in a fifth or a tenth exactly because the overtones clash, and they learn how to time/count the beats so that the amount of clash is the right amount of clash for equal temperament. Nowadays it seems like lots of tuners use mechanical aids to get the fundamentals of each string to match the frequency determined by equal temperament. I've had one of each sort of tuner lately, and I cannot hear the difference when they are finished. Harpsichordists who tune their own harpsichords have electronic pitch pies that will give you the frequencies of the fundamentals for all the strings in a whole variety of temperaments.

[timothy42b]

I'm not sure I follow this. If the fundamental frequencies are correct (according to equal temperament) then the overtones have no choice but to clash. Tuners who tune by ear know that there will be beats in a fifth or a tenth exactly because the overtones clash, and they learn how to time/count the beats so that the amount of clash is the right amount of clash for equal temperament.

Okay, let me try again.  You are quite right, my explanation was confusing and mixed a couple of concepts.

Yes, equal temperament will not produce a completely clash-less chord.  A really pure chord would have each note at a ratio of simple integers.  An octave would be 2:1.  A fifth would be 3:2.  Etc. 

As you pointed out, with equal temperament we can't get those really pure integer ratios.  We get pretty close, but it's a compromise that gives the least bad combinations, and our ears are accustomed to it.  There are other compromises possible, and some tuners will use them.  They will make a few chords better and a few worse.  My understanding is that historically harpischordists tuned their own instruments, and just selected or wrote music that didn't have the bad chords for that key signature. 

But that wasn't what I intended to convey.  There are no sine waves on a piano.  Every note has a fundamental and a set of overtones.  On a wind instrument like flute or trombone, that set of overtones will be simple interval ratios.  There will be a fundamental, and a higher overtone 2 x fundamental, and one 3 times, and one 4 times, etc.; this is forced by the physics.  Some of those overtones will be louder than others, that's why timbre varies.  Our ear hears the collection as one discrete pitch, usually the fundamental, but it isn't one pitch.  (you can filter out the fundamental but keep the overtones, and the ear will still hear the fundamental) 

The piano is the same in that there will be a set of overtones above every pitch.  It is different in that those overtones will never have a simple integer relationship.  The set of frequencies at which an object vibrates is determined by the stiffness and the mass.  A piano string is steel and has roughly the same stiffness as a wrench.  Drop a wrench on concrete and you won't hear anything like a sine wave.  The variation from integer ratios is called inharmonicity. 

The software tuning programs measure inharmonicity for that particular piano and provide a curve of recommended tuning frequencies for each note.  Aural tuners do basically the same thing by ear.   This takes  a lot of skill, plus you still have to set the string so it is stable. 

[brogers70]

Okay, let me try again.  You are quite right, my explanation was confusing and mixed a couple of concepts.

Yes, equal temperament will not produce a completely clash-less chord.  A really pure chord would have each note at a ratio of simple integers.  An octave would be 2:1.  A fifth would be 3:2.  Etc. 

As you pointed out, with equal temperament we can't get those really pure integer ratios.  We get pretty close, but it's a compromise that gives the least bad combinations, and our ears are accustomed to it.  There are other compromises possible, and some tuners will use them.  They will make a few chords better and a few worse.  My understanding is that historically harpischordists tuned their own instruments, and just selected or wrote music that didn't have the bad chords for that key signature. 

But that wasn't what I intended to convey.  There are no sine waves on a piano.  Every note has a fundamental and a set of overtones.  On a wind instrument like flute or trombone, that set of overtones will be simple interval ratios.  There will be a fundamental, and a higher overtone 2 x fundamental, and one 3 times, and one 4 times, etc.; this is forced by the physics.  Some of those overtones will be louder than others, that's why timbre varies.  Our ear hears the collection as one discrete pitch, usually the fundamental, but it isn't one pitch.  (you can filter out the fundamental but keep the overtones, and the ear will still hear the fundamental) 

The piano is the same in that there will be a set of overtones above every pitch.  It is different in that those overtones will never have a simple integer relationship.  The set of frequencies at which an object vibrates is determined by the stiffness and the mass.  A piano string is steel and has roughly the same stiffness as a wrench.  Drop a wrench on concrete and you won't hear anything like a sine wave.  The variation from integer ratios is called inharmonicity. 

The software tuning programs measure inharmonicity for that particular piano and provide a curve of recommended tuning frequencies for each note.  Aural tuners do basically the same thing by ear.   This takes  a lot of skill, plus you still have to set the string so it is stable.

Ah, thanks, this is very interesting, and there's a nice rabbit hole on inharmonicity with many entrances on the internet.

[Bob]

Trying to figure out my own original post....


There would/must always be a certain temperament I don't think you can ever get away from that. 

If you're looking for specific Hertz for pitches, it's probably equal temperament based off A 440.  And then this, yes.
https://pages.mtu.edu/~suits/notefreqs.html

And then I was still thinking about scales, so...
The tonic would be the A 440  whatever-tonic-is-in-the-key-you're-in note.  And then the scale above that... Although still equal temperament there, except for any simulataneous sounds... They would be tuned to the lowest pitch... Unless it's not possible with the music, ex. the melody staying in tune with itself and then adding a lower pitch.

And then I suppose if it was a string instrument... You'd probably tune the 5th with each other.  If you knew you were only playing in one key, you could lower the M3.  And then it's whatever tuning system that is, but the original tonic note would come from equal temperment I guess?.... For A being 440...?

And then there's reality... Most people aren't going to notice either way.  And the instrument might not be that perfect either.

And yep, for the other comments about piano tuning being a little different, overtones, and things not being perfect in the real world.  Although, other instruments are going to have their own imperfections.  Nothing would be able to produce mathematically perfect pitches, overtones, etc.  Something is always going to be off somewhere, somehow.