Inharmonicity

[Enthalpy]

Hello everybody and everyone!

I propose to discuss here the acoustic inharmonicity, a theory very well know, especially for musical strings.

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Piano and cimbalom string choirs have different lengths depending on the note height, and the bridge is consequently skewed. The string length varies smoothly on a cimbalom, including within a choir, while piano luthiers painstakingly carve the bridge to have the same string length within a choir and let it jump between the choirs.

I claim this results from string inharmonicity. The diameter of a string makes it stiff, which shifts the partials above the harmonic frequency, as is known for decades.

This theory is simple, it has some experimental support, so it has been misused thousand times. At choirs it seems useful: if strings of different lengths have their fundamental in tune, the partials can't be aligned, and since the amplitudes are similar, the beats should be strong and easily perceived within a single choir. Numerically, the differences range from few Hz to many dozens depending on the rank, and string instruments typically have partials stronger than the fundamental.

To make an auditive opinion, I wrote Choir.cpp, while ChoirMake.txt lets paste commands to feed Choir.exe that outputs Choir.wav. This zip also contains .wav of four notes, 2s with identical lengths in a choir and 2s with lengths varying as tempted by the bridge skew.

InharmonicChoir.zip

The beat is clearly heard and sounds badly within the long sustain of a piano note. It looks like a good explanation for the efforts of piano luthiers.

Would cimbaloms, Hackbrett, hammered dulcimers et al benefit from the piano design? Maybe. Their presently unstable tuning must overshadow all subtle effects, but this may improve in the future, as I described metal frames on July 08, 2019 and July 21, 2019 and in nearby messages, so if they become as stable as a piano, choirs with uniform length may be te next improvement.

Marc Schaefer, aka Enthalpy

[Enthalpy]

Brass wind instrument have pedal notes. Bad and normally unused on the trumpet, not good on the tenor trombone, excellent and commonly used on the tuba and bass trombone, whose larger bore logically stabilize the lower modes.

The first mode is not an octave below the second mode on my tuba. It's a fifth, nicely accurate. Wiki gives an exhilarating explanation for that:
wikipedia

Quote

"Some tubas have a strong and useful resonance that is not in the well-known harmonic series. For example, most large Bb tubas have a strong resonance at low Eb (Eb1, 39 Hz), which is between the fundamental and the second harmonic. These alternative resonances are often known as false tones or privileged tones.

The most convincing explanation for false tones is that the horn is acting as a "third of a pipe" rather than as a half-pipe. The bell remains an anti-node, but there would then be a node one-third of the way back to the mouthpiece. If so, it seems that the fundamental would be missing entirely, and would only be inferred from the overtones. However, the node and the antinode collide in the same spot and cancel out the fundamental."

It should be worth reminding that the "harmonic series" happens in cylindrical pipes open at both ends or closed at both ends, and nearly so in open conical pipes. The alphorn, cornetto, serpent, ophicleide are conical. The trumpet, flugelhorn, trombone, saxhorn, tuba, no usual brass is conical - just have a look at one.

Modes 2 and over are aligned because the luthier adjusts the flare for it with much effort. The valves or slide need some cylindrical portion so the flare can't adjust the first mode to the octave. The "alternative resonance, false tone, or privileged tone" is the fully normal first resonance mode, just not at the height some people imagine.

This resonance is fuzzy on a tuba but rather well tuned. I suspect the luthier carefully adjusts it to a useful and accessible height: the fifth combines nicely with the valves, and the harmonics 3, 6, 9... lock into modes 4, 8, 12... to stabilize the note and sound better.

I didn't try if already the second mode of a baroque trumpet is out of tune, nor where the first mode is on a trombone, nor if a flugelhorn could imitate a tuba. Whether musicians impose with the lips a height that is not the first mode, and merely the harmonics locked in higher resonance modes give some stability?

Marc Schaefer, aka Enthalpy

Edited May 10, 2020 by Enthalpy

[Enthalpy]

21 hours ago, Enthalpy said:

The first mode is [...] a fifth [below the second mode], nicely accurate. [...]
I suspect the luthier carefully adjusts [this resonance] to a useful and accessible height [...]

As it looks, I botched that part. Diagrams of the input impedance versus the frequency are available on the Web for the trumpet and trombone, starting possibly with
John Backus , The Acoustical Foundations of Music
cited among others by
hyperphysics.phy-astr.gsu.edu - newt.phys.unsw.edu.au
The tuba must behave like the trumpet and trombone, with a first mode lower than half the second mode frequency and the pedal notes, not higher.

So the useful "privileged tones" would result from some mode-locking between the sound's harmonics and the instrument's resonances, just as they are known to do with the pedal tones at the octave below.

But then, why do the harmonics 3*M lock in modes 4*N for a tone a fifth below mode 2? The harmonics 2*M could also lock into 3*N for a tone a fourth below mode 2. Lower harmonics and a less low tone should favour that.

Wiki's 39Hz is indeed an Eb, a fifth below the Bb second mode. On my tuba after 20 years, I'm not so sure.