# Are bells the only instrument whose second harmonic forms a minor interval with its fundamental?

I heard (from a Paccard church bells rep) that bells are the only instrument whose second harmonic forms a minor interval with its fundamental. Is this true?

• It's not a harmonic if it doesn't follow harmonic series. Appropriate names in this case are partials and overtones. Commented Feb 27 at 4:31
• The pitch we perceive on a "minor third" bell is the 5th partial, aka 4th harmonic, with the secondary tone being the 3rd partial. The reason we don't perceive the fundamental as the bell's pitch is that when it is struck with a hard object at the right place, the fundamental tone is inaudibly weak, just as the fundamental is relatively weak in violins, oboes, and other bright-sounding instruments. Another instrument whose apparent pitch is different from its fundamental is the Timpani (well, timpano since it's singular), sounding an octave above the fundamental. Commented Feb 28 at 17:34
• Does this minor-third-first-partial property apply to church bells only, or tubular bells as well?
– John
Commented Sep 9 at 2:43

In short the question boils down to: are there any other instruments able to produce a spectrum which does not follow the harmonic series, neglecting inharmonicity.

Well, any instrument whose oscillator is more than unidimensional (most "standard" instruments fall in that 1D category: the air column is 1D in winds, strings are 1D...) is in theory able to produce overtones that do not follow the harmonic series. As you stated, bells can (they are basically a surface as the oscillator (not the resonator), hence 2D) but so can every instrument with a membrane (drums, tabla, steel drum...). Now, it is true that the envelopes of these are generally very different from bells, but even if none is tuned like that now, it is theoretically possible for all this family of instruments.

• Even some "unidimensional" objects produce non-harmonic overtones. For example, anything involving the vibration of a "stiff" object (e.g., xylophone, glockenspiel, tubular bells, kalimba) produce harmonics that are not in a strict harmonic series. Commented Feb 27 at 21:29
• @MichaelSeifert But if I'm not mistaken, all of your examples are 2D... Tubular bells are cylinder (not an air column) and the others are surfaces with length and width. All of this is of course forgetting about inharmonicity.
– Tom
Commented Feb 27 at 21:39
• The most important vibrational modes are along the length of the bar/tube/etc; there's minimal flexing or twisting in these cases. Mathematically, this means that the vibrations are effectively 1D, since the transverse dimensions don't come into play. It's similar to how an air column is really a 3D volume of air, but in the modes that are important for instruments, the pressure only varies along the length of the column and not across the length of the tube. Commented Feb 28 at 0:06
• @MichaelSeifert Yes, but the approximation of unidimensionality does not hold as well as for a string or a normal air column. A thin plate will have significant flexing and torsion, depending how are the boundaries conditions. Thin cylinders can also exhibits modes along their length and in their sections, none of them negligible
– Tom
Commented Feb 28 at 6:40
• @MichaelSeifert I tend to agree with Tom; for instance, while keyboard percussion instruments can be simplified as unidimensional, their bars are still quite bidimensional (and, in fact, three dimensional), and the second dimension can affect a lot: given a large enough bar, and having 4 parallel points of suspension (where the strings normally are), their horizontal position (the distance from the vertical axis) can change how the body can actually vibrate just as much as the vertical distance does. Commented Feb 29 at 1:27

There are "minor third bells" (e.g., church bells) and "major third bells", thus instruments can be tuned to have these different properties.

The correct terminology in the context of bells is "prime" and "tierce" (not "fundamental" and "second harmonic"; as user1079505 notes, "It's not a harmonic if it doesn't follow harmonic series.").

Table 6.2 of Sethares, Tuning, Timbre, Spectrum, Scale (p. 117) gives the partials:

Name of Partial Ideal Minor Third Bell Measured Bell Ideal Major Third Bell
hum 0.5 0.5 0.5
prime 1.0 1.0 1.0
tierce 1.2 1.19 1.25
quint 1.5 1.56 1.5
nominal 2.0 2.0 2.0
deciem 2.5 2.51 2.5
undeciem 2.61 2.66 2.95
duodeciem 3.0 3.01 3.25
upper octave 4.0 4.1 4.0

Ibid. p. 116:

Traditional church bells tuned this way are called “minor third” bells because of the interval 1.2, which is exactly the just minor third 6/5. Bell makers have recently figured out how to shape a bell in which the tierce becomes 1.25, which is the just major third 5/4. These are called “major third” bells.

For example, this bell is a very good minor third bell:

Name of Partial Ideal Minor Third Bell Ratio Measured Measured Ratio
hum 0.5 144 Hz (Re3) = -41,0 dB 0.51
prime 1.0 284 Hz (Do♯4) = -37,4 dB 1.0
tierce 1.2 343 Hz (Fa4) = -43,2 dB 1.21
quint 1.5 431 Hz (La4) = -55,7 dB 1.52
nominal 2.0 576 Hz (Re5) = -39,1 dB 2.03
deciem 2.5 717 Hz (Fa5) = -50,1 dB 2.52
undeciem 2.61 744 Hz (Fa♯5) = -53,5 dB 2.62
duodeciem 3.0 858 Hz (La5) = -40,8 dB 3.02
upper octave 4.0 1181 Hz (Re6) = -45,9 dB 4.16
• This does not really answer the question "Are bells the only instruments..."
– Tom
Commented Feb 29 at 18:30
• @Tom It does make clear that bells can be tuned not to be minor third bells. Analogously, other instruments could be tuned to have different partials. Commented Feb 29 at 18:34
• This analogy does not quite hold... It is not because some can be tuned not to be minor third that other instrument can. And the question was: Are bells the only instrument that [...], Which you answer by: some bells don't. But that does not tell about the "only instrument" at all. Maybe I'm nitpicking, I like your answer but I just feel it is not really connected..
– Tom
Commented Feb 29 at 21:55

The second lowest overtone of a theoretical drumhead is 2.136x the fundamental mode (+1314 cents) (here, here) which is within 3 cents of a just minor ninth.

Also, does a synthesizer count? It feels like cheating, but it is an instrument, and it can be made to produce any overtones.

• Yes, a synth doesn't count, since as you say it just calculates the vibrational pattern of the sum of specified input frequencies. Commented Mar 1 at 15:39