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Originally Posted by An Old Square
Because of inharmonicity in piano wires, NO "harmonic" above the fundamental can be considered a true harmonic.

There is a reason, besides nomenclature, to consider the fundamental also subject to the same physical mechanism that causes inharmonicity in the higher partials. That mechanism is the role of stiffness of the wire in determining pitch.

The physics of vibrating strings can be analyzed in the ideal case where the string has no stiffness to bending, but does have elastic resistance to being stretched. In this case the pitch is determined only by the tension in the string, the spring constant of the string, and the mass of the string. The physics of vibrating strings predicts that in such an idealized case, the partials are indeed all whole number multiples of the fundamental. This situation is approached by harpsichord strings where the strings are so thin that resistance to bending is almost inconsequential. And in fact if you try to measure the inharmonicity of the longer and thinner harpsichord strings you will find it so low as to be indistinguishable from zero.

Now if we take that idealized model of vibrating strings with no resistance to bending and simply add in that stiffness, the theory of vibrating strings can then calculate the resulting pitch of the fundamental and all the higher partials. The result is, given that nothing else changes except the stiffness, the pitch rises for all the partials and the fundamental as well. That is, the fundamental pitch is also affected by the inharmonicity constant. But the amount by which the pitch is affected is proportional to the square of the partial number. So if the fundamental is partial 1, followed by partials 2, 3, 4, etc., the pitch due to the inharmonicity constant goes up by a number of cents proportional to 1, 4, 9, 16, etc. That is, the higher partials are affected much more by stiffness than the lower partials and the fundamental. This makes sense too because if you picture a string vibrating at a higher partial you can see the string is being bent at more places (every node) whereas the fundamental is only being bent very little. So resistance to bending is going to affect higher partials more.

To put this in mathematical terms, the offset in cents due to inharmonicity (string stiffness) is given by:

Offset = Ihcon * p^2

where p is the partial number and Ihcon is the inharmonicity constant. The offset is with respect to the harmonic frequency, the whole number multiple of the idealized fundamental, which I will call f0. What is interesting is that the actual fundamental that we would measure in such a situation is not f0. It is f0 offset by Ihcon * p^2.


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Originally Posted by David Boyce
This is an interesting discussion. Terminology seems somewhat variable.

Indeed!

I just (*gasp*) Googled "what is the difference between harmonics and partials?".

Even from scholarly sources (both musical and scientific), the range of definitions, and nuances and subtleties between very similar definitions, seems more like a Bayesian distribution* of every possible definition than anything likely to settle this one.

Long as it's tuned right, the definitions matter not.

Reminds me of the first time I tuned for James Taylor. Nervous youngster trying to nail it. I was on C8, last note, but not liking what I'd done, so I kept trying, while the clock was ticking.

Out of the shadows of a corner steps James, who'd been listening to me struggle. No clue how long he'd been hanging out listening (ie waiting for me to GET OFF THE STAGE).

He walked up to the piano, looked at me, and said "Don't worry about it, it's fine. It's a C, it sounds fine to me, and I doubt it's getting played today anyway."

(Not my usual clientele attitude tho, lol.)


*and yes, I may have used that term just cuz it makes me sound way smarter than I actually am

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Old Square, your shorthand was not the problem. I understood well enough that those numbers were referring to steps in the diatonic scale.

But all tuning nomenclature that I've ever seen, is based on the convention of the fundamental being the first partial, in order to get the rest of the series correct.

To use your practical example of C3: I have never heard or read of G4 being the second partial of C3, as you state it to be. When tuning C2-C3 as a 6:3 octave, we compare beat speeds at G4. This means that G4 is the 6th partial of C2 and the 3rd partial of C3.

Reblitz, again (p. 214): "For example, to tune a "4:2" octave, you tune the 4th partial of the lower note to the 2nd partial of the upper one. To tune a "6:3" octave, you tune the 6th partial of the lower note to the 3rd partial of the upper one."

All of this only works if the fundamental is the first partial.

The German wikipedia page on this subject (https://de.wikipedia.org/wiki/Oberton) says the same:
"Der tiefste Teilton wird Grundton genannt und bestimmt in der Regel die wahrgenommene Tonhöhe."
(The lowest partial is called fundamental, and as a rule, determines the perceived pitch.)

And the German wikipedia page on inharmonicity (https://de.wikipedia.org/wiki/Inharmonizit%C3%A4t) also says the same in the section on vibrating strings:
"Die Schwingung ihrer gesamten Länge erzeugt den Grundton oder 1. Teilton, die jeweiligen Abschnitte die Obertöne, das heißt den 2. Teilton, den 3. Teilton, den 4. Teilton usw."
(Vibration of its entire length generates the fundamental or first partial, [while] the respective sections generate the overtones, i.e. the second, third and fourth partial, etc.)

I sense that my interjected queries are detracting from the original question. Apologies to the thread starter. I'll leave it at this.

[Edit: Sorry, only saw your last post now. Agreed, tuning it right is more important that the definitions...]

Last edited by Mark R.; 11/21/21 03:32 PM. Reason: given in post

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Ah, got it.

You are quite right and not at all wrong, given that all systems *I also * have ever seen, that explain what coincident partials are, require that, for example, a 4:2 octave (ie as in tuning width) means that the 4th (something) of one note (always meaning the double octave *pitch* above the fundamental) and the 2nd (something) of another note (always meaning the octave *pitch* above the fundamental) are tuned beatless; same for 2:1 or 6:3 or any other combination indicating coincident partials.

Were any system of description to hew to what *I* said in this thread, lordy, what a mess grasping what is said quickly and simply would be!

I also use the same system of (number)(colon)(number) to reference two (or more) *tones* that coincide at the same (or nearly the same) frequency, and I also call and have always called that combo a "coincident partial"; exactly like you, and everyone else.

Without all the picky "but the fundamental, she isn't a partial, she's a harmonic!" stuff I've dumped in this thread. I just say, tune a 6:3 octave.

In actual practice (communicating with others, except for here), I have just accepted the convention as you've described, and used the terms as you've described.

So why oh why have I apparently gone off the deep end here, today? wink

Because, there was a fairly specific point being made about terms and nomenclature on a granular level. Not the macro level of how *tuning works*, but the granular level of what *words mean*.

And, I will insist, that if one is being very precise with terminology, the entire *pedagogic and expert world* made a decision, to opt for simplicity and convenience in terms *long ago*, that is just flat out *wrong*. And that has become a convention.

If one insists (as I do) that it IS perfectly OK to conform to descriptive conventions that may be somewhat incorrect if one takes a microscope to semantics of terms (the double octave *tone* contained in a note is NOT the "4th partial", because it cannot be, because the fundamental is NOT the first partial contained in the note, because, good grief, what is the fundamental supposed to be a PART of? The fundamental is created by the strings WHOLE length. Is that whole string a partial itself of an imaginary string twice it's length? In some other dimension?)

I am probably a victim of the memory of a PTG Examiner, bellowing at a class of students "If the fundamental is a partial, WHAT IS IT A PART OF, THEN, HMMM?"

I am being a linguistic stickler here in insisting (in the context ONLY of examining terminology) that using the word "partial" to refer to the vibrations produced by WHOLE strings, is fine and OK in everyday communications (eg, I tuned a lot of 8:4s in that bass, etc); and I am certainly not on a mission to correct a million errors long etched in stone everywhere; but dang if I'll agree all mammals are dogs, even if all dogs are mammals.

Anyway, this has been fun!

And I swear and aver and affirm I will never ever make this point here again.

The world got it wrong long ago as far as linguistic precision goes, it cannot be undone, and the fundamental is a partial.

Don Quixote hereby retires from tilting at that windmill.

smile

*But if anyone calls the fundamental an OVERTONE, that's a bridge too far and my knives are coming out!!!

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I've kept out of this as I can never decide which terms to use for all this.
However if I had been there:quote
I am probably a victim of the memory of a PTG Examiner, bellowing at a class of students "If the fundamental is a partial, WHAT IS IT A PART OF, THEN, HMMM?" Unquote.
I would have replied " it's a part of the sound" , maybe, if I'd had the nerve, even bellowing the word "sound".
For your esteemed examiner was assuming he knew that the word partial was referring to the fundamental. But it's equally valid ( or maybe more so) to hold that the word partial is referring to the set of frequencies that make up the overall "sound" of the note.
So he was guilty of narrow vision and worse, having the nerve to bellow his opinion at a group of students trying to learn the complex issue.
Nick

Last edited by N W; 11/21/21 06:27 PM.

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Originally Posted by N W
I've kept out of this as I can never decide which terms to use for all this.
However if I had been there:quote
I am probably a victim of the memory of a PTG Examiner, bellowing at a class of students "If the fundamental is a partial, WHAT IS IT A PART OF, THEN, HMMM?" Unquote.
I would have replied " it's a part of the sound" , maybe, if I'd had the nerve, even bellowing the word "sound".
For your esteemed examiner was assuming he knew that the word partial was referring to the fundamental. But it's equally valid ( or maybe more so) to hold that the word partial is referring to the set of frequencies that make up the overall "sound" of the note.
So he was guilty of narrow vision and worse, having the nerve to bellow his opinion at a group of students trying to learn the complex issue.
Nick

Hmmm... interesting.

Replace "sound" with "the whole waveform of the note", and he might have actually stopped for a second, and then agreed that was a quite valid way of looking at it, and thanked you in front of the class for bringing a new perspective he had not considered. He would then be forced to include THAT perspective in that discussion - forevermore - along with his.

(But "sound" is FAR too layman-ish and general a word to have worked, lol, and he would have sentenced you to pitch raising a spinet the rest of the day, had you actually bellowed "sound" at him. And not the nice restored 5 cent flat Acro, the just-in non-functioning minor third flat Aeolian.)

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What may even be more interesting (repeat...MAY) is that we do not actually "hear" much, if any at all, of the fundamental. Our brain "synthesizes" (or interprets) the sound of the fundamental as a result of the combined upper partial series. In fact, all "hearing" is in fact the brain's interpretation (and memorization) of combinations of partials, and relative amplitudes of said partials (harmonic or inharmonic).

Also, interestingly, years ago Dr. Al Sanderson (of Accutuner fame) mentioned that he had actually measured NEGATIVE inharmonicity in one brand of piano. Any guesses as to what piano he was talking about? (He also called it: "one weird piano" or possibly "one strange piano"...I don't remember the precise wording now 35 years later).

Peter Grey Piano Doctor

Last edited by P W Grey; 11/21/21 08:07 PM.

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Originally Posted by P W Grey
What may even be more interesting (repeat...MAY) is that we do not actually "hear" much, if any at all, of the fundamental. Our brain "synthesizes" (or interprets) the sound of the fundamental as a result of the combined upper partial series. In fact, all "hearing" is in fact the brain's interpretation (and memorization) of combinations of partials, and relative amplitudes of said partials (harmonic or inharmonic).

Also, interestingly, years ago Dr. Al Sanderson (of Accutuner fame) mentioned that he had actually measured NEGATIVE inharmonicity in one brand of piano. Any guesses as to what piano he was talking about? (He also called it: "one weird piano" or possibly "one strange piano"...I don't remember the precise wording now 35 years later).

Peter Grey Piano Doctor

+1!

I remember my first synthesizer (love them too!) at 11, 1970.

Scientist nerd LANL physicist dad made it for me outa Radio Shack parts (to my specs).

It had a single pure sine wave oscillator, from 1hz to 20Khz, and a 128 note sequencer, had to use them all.

Same length notes only, but could vary speed between 1 note per minute to 1000 notes per second.

Vividly recall how shocked I was at how weak and quiet pure sine waves became in the lower frequencies.

Later synths: add a few higher harmonics using square etc waveforms, and the fundamental suddenly *appears* to become louder - even when very low.

That's when I began to suspect exactly what you wrote and have been interested in this ever since.

Thanks!

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Originally Posted by P W Grey
guesses as to what piano he was talking about?
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Yamaha G3 / Pianoscope

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Originally Posted by P W Grey
What may even be more interesting (repeat...MAY) is that we do not actually "hear" much, if any at all, of the fundamental. Our brain "synthesizes" (or interprets) the sound of the fundamental as a result of the combined upper partial series. In fact, all "hearing" is in fact the brain's interpretation (and memorization) of combinations of partials, and relative amplitudes of said partials (harmonic or inharmonic).

Also, interestingly, years ago Dr. Al Sanderson (of Accutuner fame) mentioned that he had actually measured NEGATIVE inharmonicity in one brand of piano. Any guesses as to what piano he was talking about? (He also called it: "one weird piano" or possibly "one strange piano"...I don't remember the precise wording now 35 years later).

Peter Grey Piano Doctor

There is a nice photo of PianoMeter's frequency spectrum graph for note A2 on my upright piano, on this page of my website https://www.davidboyce.co.uk/electronic-tuning.php. It shows the low power of the fundamental (first partial) , compared to the second partial, A3 and third partial E4.

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This thread reminds me of an experiment that was done when I was young. I'm searching my old IMIT records for details but basically it was that in those days the telephone system had a bandwith to only cope with speech. Charcoal microphones..anyone else remember them? Anyway this scientist musician wrote and recorded a piece of music to be sent over the telephone wires, constructed in such a way that bass notes could be heard even though they could not possibly have been transmitted and carried by the phone system. I just can't remember whether it was in UK or USA.

The brain definitely makes calculations from what it hears and presents us with something that "makes sense".
On a small upright there is often no fundamental from c3 down. But, although we do "hear" the fundamental that isn't there, it's not quite the same in texture to me....hence the happy, big smile that appears on my face when I hit a low note on a lovely big Bosie...ahhhhh real bass! smile


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Originally Posted by An Old Square
[te
I am probably a victim of the memory of a PTG Examiner, bellowing at a class of students "If the fundamental is a partial, WHAT IS IT A PART OF, THEN, HMMM?" Unquote.
I

Hmmm... interesting.

Replace "sound" with "the whole waveform of the note", and he might have actually stopped for a second, and then agreed that was a quite valid way of looking at it, and thanked you in front of the class for bringing a new perspective he had not considered. He would then be forced to include THAT perspective in that discussion - forevermore - along with his.

(But "sound" is FAR too layman-ish and general a word to have worked, lol, and he would have sentenced you to pitch raising a spinet the rest of the day, had you actually bellowed "sound" at him. And not the nice restored 5 cent flat Acro, the just-in non-functioning minor third flat Aeolian.)
smile smile
Actually, having thought about it for longer, I would want him to accept my definition as preferable, in fact correct, not equal!
What about this?
If we want to describe 880 vibrations as the first "partial" of 440 vibrations....in what way can that make sense?
880 exists independently from 440 (and so on up the set). So 880 is partnering 440...in a sense when we go to A5 from A4 what we have done is remove the slower vibrations. So the note A5 is A4 without the 440 content.
To put it another way. Take a mathematical multiplication table, which seems to me to be analogous, and to keep it simple let's use the 1 times table.
1*1=1,1*2=2 and so on.
The table represents the whole waveform of the sound.
So 1*1 is obviously the fundamental of the system.
Now 1*2=2
In what way can we say 2 represents part (is a partial of) of the 1?
2 is part of the "table" not the fundamental, as is 1*1 in fact;
As soon as there is more than one pitch vibrating in a string, we have a system, none of the parts are part of the fundamental, they are part of the whole waveform of the sound. The parts necessarily include the fundamental.

What do you think?

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I think you're right.

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Harmonics are integer multiples of the fundamental frequency. 1 is an integer. 440 multiplied by 1 is 440.

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Partial: any of the component tones of a single musical sound, including both those that belong to the harmonic series of the sound and those that do not.

https://www.collinsdictionary.com/dictionary/english/partial


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It is wonderful to see here a rational discussion regarding 'words' and the difficulty they present when attempting to communicate a precise concept. Thank you to all who have contributed.


I like to differentiate the use of the words 'partial' and 'harmonic', whether 'correctly' or not, with 'partial' referring to those frequencies as defined in Withindale's reply above, - and 'harmonics' strictly limited to precise integer based multiples of the lowest frequency.

I do this because, in Radio Frequency amplifier design, I am concerned with knowing the precise multiples of my transmitted frequency that may cause interference with other services. These multiples do not exhibit inharmonicity since the vibrating source has no fixed nodes in space.

I also need to know the approximate stretch of each partial on my fixed node piano strings in order to best render a reasonable temperament with as much consonance as is possible for that instrument due to the interference of the strings with the other strings.

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Originally Posted by David Boyce
Harmonics are integer multiples of the fundamental frequency. 1 is an integer. 440 multiplied by 1 is 440.


Zero is also an integer :-)

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Zero is an integer, and an important integer. It defines unity.

1^0=1
2^0=1, 2^1=2, 2^2=4
3^0=1
4^0=1
...

This implies that the frequency of the first harmonic, if you will, is the fundamental frequency.

Great stuff people. Continue.

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Originally Posted by prout
Zero is an integer, and an important integer. It defines unity.

1^0=1
2^0=1, 2^1=2, 2^2=4
3^0=1
4^0=1
...

This implies that the frequency of the first harmonic, if you will, is the fundamental frequency.

Great stuff people. Continue.

Let's also not forget subharmonics! These also contribute to the perception of the (missing) fundamental. For example, the difference tone of the 2nd and 3rd partials is a subharmonic one octave below the fundamental. These are produced in the ear by the non-linear way the eardrum works (if I remember correctly) and the difference frequencies are easier to hear than the summation ones. Fun to listen out for!

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