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I would like to understand why when using PianoMeter's spectrograph (and presumably with any ETD) the second partial remains very strong even as the fundamental decays? I was checking in the temperament zone.

Let me give you an example from another area: many people think that the singer sings an octave higher. Apparently, this is the physics of this process, when the second (octave) partial bulges out strongly in level. In My Humble Opinion.

This looks like an interesting observation which my simple physics is not explaining. By second partial do you mean the second harmonic, in other words if you play a C, the first harmonic is octave C and the second harmonic is G. The frequency of G is half way between two C's and could that be a factor ? Could there be an explanation in the attenuation coefficient being a function of frequency?
Does this observation hold true for a wide range of notes from very bottom to the top of the piano or are some types of strings more affected than others?

keff, if first partial is C, second is also C and third is G above that

Thanks, my background is physics (but a long time ago) and the nomenclature I use is fundamental, first harmonic etc.. In your nomenclature the second partial consists of one complete wavelength and perhaps less energy is required to sustain a full wavelength rather than one half wave.

The second partial is the tone that is produced by a piano string which happens to be very close in pitch to the second harmonic, but not exactly the second harmonic. Piano strings do not produce any second harmonic, or indeed any harmonic, only partials. The distinction is that harmonics are, by definition, exact multiples of the fundamental frequency. Partials are the actual tones that are present.

As for the reason that partials decay at different rates, that is because each partial is ringing independently of all the others, and the mechanism for energy transfer out of the string is dependent on the frequency. At some frequencies the bridge appears more stiff than at other frequencies. Therefore those frequencies persist longer.

Are you talking about a single string or a full unison?

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Regardless of harmonic or inharmonic overtones, there are two distinct sets of nomenclature:

Physicists mostly use the system of overtones: fundamental, first harmonic/overtone (octave) , second harmonic/overtone (octave+fifth), etc.
Tuners mostly use the system of partials: first partial, second partial (octave), third partial (octave+fifth).

In other words, the physicist's fundamental corresponds to the tuner's first partial, the first overtone/harmonic corresponds to the second partial, the second overtone/harmonic corresponds to the third partial.


This is a nomenclature issue.

Whether the overtones or partials are perfectly harmonic or inharmonic, is another matter and depends on the sound generator, e.g. pipes (vibrating air column) are harmonic, while strings are inharmonic because of material stiffness.

Thanks Mark R, that is what I wanted to clarify before starting to think about the question put by the OP.

The following Youtube videos provide commentaries on the spectral analysis of piano tones. There are seven parts in all.

Physicists use both the terms - harmonic and partial - very carefully depending on the situation. See Fundamentals of Musical Acoustics by AH Benade 5.6 page 63.

"... the partials of a guitar string have frequencies which are very nearly but not exactly harmonics of the frequency of the first (lowest) partial."

One reason why the decay rates of different partials differ is the variance in the impedance of the soundboard. You can find calculations on this on page 294 of this paper ( https://www.jstage.jst.go.jp/article/ast1980/10/5/10_5_289/_pdf )

Another reason - if you are analysing the sound of a unison - is the micro mistuning between the unison strings. An observed fast decay of a partial can simply be a slow beat.

By the same token, instrumentalists like violinists and guitarists speak of harmonics (e.g. when creating flageolet overtones by lightly touching the string) even though they aren't strictly harmonic overtones.

My point was that before we even get into harmonic vs. inharmonic, we clarify the counting systems.
... the partial system starts count at the fundamental, while
... the overtone / harmonic system starts count at the first overtone, not the fundamental.

Harmonics begin with the fundamental. The fundamental IS the first harmonic. It is NOT a partial (it is the whole string).

Partials begin with the octave harmonic. The octave harmonic is the first partial (the first not-whole-string harmonic) but the second harmonic.

Overtones are more of an informal laypersons term and best avoided in technical discussions but are equivalent to partials.

Harmonic series from the first harmonic to the quadruple octave harmonic:
1
1
5
1
3
5
7
1
2
3
4
5
6
7
7
1

Partial series from the first partial to the quadruple octave partial:
1
5
1
3
5
7
1
2
3
4
5
6
7
7
1

Best not to refer to an "overtone series" at all.

Harmonics/partials from the quadruple octave to the quintuple octave are wildly divergent from the musical scale, there being 16 pitches within that octave (some of whom do not correspond to any musically used pitch), and at quadruple octave plus inharmonicity, depending on the length/diameter/tension of the wire, may be so wildly sharp that even the consonant harmonics such as the quadruple octave fifth are too sharp to recognized as such.

The best place to explore the quad to quint harmonics is in the lowest plain tenor wires on a bright 9' grand.

I'm open to debate about which exactly is the first harmonic.

But with due respect to your immense experience, your statement on partials is wrong.
See, for example, diagram 6-2, p. 204, Piano Servicing, Tuning, and Rebuilding, Arthur A. Reblitz (2nd ed., 1993)

The correct partial series does include the fundamental, as follows (steps of the scale given in brackets):
P1 = whole string = fundamental (1).
P2 = half string = octave (1).
P3 = one-third string = 12th (5).
P4 = one quarter string = double octave (1).
P5 = one fifth string = double octave plus M3 (3).
etc.

The octave partial series consists of partials 1, 2, 4, 8, 16 etc. - an octave always involves a doubling of partial number. After all, why else would we speak of a 2:1, 4:2 or 6:3 octave?

But in your list, the octave partials are listed as entries no. 1, 3, 7, 15.

I *do* apologize, I was *very* lazy in my post with the numbering system. I used a shorthand incorrectly assuming others would recognize it. The numbers refer just to each harmonic's/partial's place in the diatonic scale, without reference to major or minor, for brevity, but obviously not clarity. I used this system as a way to teach my tuning students an easy way to remember the entire harmonic series to the quad octave in a way that could be rhythmically chanted and memorized for life in minutes.

Thus, as a chant: One one five one three five seven one two three four five six seven seven one.

(However, Mr. Reblitz in his book, which was indeed my primary textbook at tech school, is also guilty of some nomenclatural laziness. This was pointed out at the time by our tuning teacher, Lawrence T. Goetsch, who was at the time a PTG Examiner, and a friend of Mr. Reblitz. To call the whole string fundamental harmonic a partial is just silly on the face of it. However I understand why he did that, and in the grand scheme, it isn't important.)

Written out fully using C3 to start, with my shorthand on the left of each note:

1 C3 Fundamental First harmonic, and the only non-partial harmonic

1 C4 Octave harmonic First partial and the second harmonic

5 G4 Octave fifth harmonic Second partial and the third harmonic

1 C5 Double octave harmonic Third partial and the fourth harmonic

3 E5 Double octave major third harmonic Fourth partial and the fifth harmonic

5 G5 Double octave fifth harmonic Fifth partial and the sixth harmonic

7 B-flat5 Double octave minor seventh harmonic Sixth partial and the seventh harmonic

1 C6 Triple octave harmonic Seventh partial and the eighth harmonic

2 D6 Triple octave major second harmonic Eighth partial and the ninth harmonic

3 E6 Triple octave major third harmonic Ninth partial and the tenth harmonic

4 F6 Triple octave perfect fourth harmonic Tenth partial and the eleventh harmonic

5 G6 Triple octave perfect fifth harmonic Eleventh partial and the twelfth harmonic

6 A6 Triple octave major sixth harmonic Twelfth partial and the the thirteenth harmonic

7 B-flat6 Triple octave minor seventh harmonic Thirteenth partial and the fourteenth harmonic

7 B6 Triple octave major seventh harmonic Fourteenth partial and the fifteenth harmonic

1 C7 Quadruple octave harmonic Fifteenth partial and the sixteenth harmonic


By substituting "harmonic" for where Mr. Reblitz "partial", and getting my teaching shorthand, you'll see that Reblitz and I are in total agreement.

Finally, addressing a point made elsewhere in the thread, re the difference between the nomenclature in physics and as used by tuners: I have two PhD physicists in my immediate family, who have worked on projects for LANL, Sandia, JPL, and NASA, which involved harmonics, as defined in physics. In physics, a harmonic MUST be an EXACT fractional ratio of some sort. Because of inharmonicity in piano wires, NO "harmonic" above the fundamental can be considered a true harmonic. In real world physics projects, a tiny mathematical error (such as ignoring a non-precise harmonic ratio) can result in death, destruction, mayhem, and millions or billions in losses. Since no such dire consequences exist in the world of tuning, if tuners get their nomenclature mixed up or confused or reversed or transposed, the stakes involved in being precise are many orders of magnitude less serious, lol.

I do appreciate you keeping me honest and making me stop being so lazy and unclear in my posts! Bad habit!!!

This is an interesting discussion. Terminology seems somewhat variable. On "Harmonic", Wikipedia says

But under a Subheading "Terminology" it notes

freemusicdictioary.com says of Partials:

In piano strings, I have understood Harmonics to refer to the mathematical integer multiples of the fundamental (including the multiple 1, the fundamental), and Partials to refer to actual frequencies produced by the strings moving in parts (including the whole length, the 1st partial).

Frank, I was referring to a single strings and for the sake of clarity the fundamental and partial are e.g. A4 and A5 where A5 has is markedly greater sustain long after A4 has decayed so the ETD is displaying it as zero amplitude.